Displaying 1-10 of 14 results found.
G.f. satisfies A(x) = (1 + x^2)*(1 + x*A(x)^2).
+0
10
1, 1, 3, 8, 24, 80, 278, 997, 3670, 13782, 52588, 203314, 794726, 3135540, 12470444, 49942305, 201233170, 815205699, 3318291966, 13565162636, 55669063762, 229257178198, 947142023262, 3924380904498, 16303716754884, 67899954924360, 283425070356740, 1185551594834910
COMMENTS
More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).
FORMULA
G.f. satisfies:
(1) A(x) = (1 - sqrt(1 - 4*x*(1+x^2)^2)) / (2*x*(1+x^2)).
(2) A(x) = exp( Sum_{n>=1} x^n/n * A(x)^n * [Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^(2*k)] ).
(3) A(x) = exp( Sum_{n>=1} (1-x/A(x)^2)^(2*n+1)*[Sum_{k>=0} C(n+k,k)^2*x^k/A(x)^(2*k) )] * x^n*A(x)^n/n ).
(4) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A200717. (5) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A200717. Recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) - (n+1)*a(n-2) + 6*(2*n-5)*a(n-3) + 6*(2*n-9)*a(n-5) + 2*(2*n-13)*a(n-7). - Vaclav Kotesovec, Aug 19 2013 a(n) ~ c*d^n/n^(3/2), where d = 4.41997678... is the root of the equation -4-8*d^2-4*d^4+d^5=0 and c = sqrt(d*(8 + 16*d^2 + 8*d^4 + 3*d^5 + d^7) / (Pi*(1 + d^2)^3))/4 = 0.648259186485429075561822659694489853... - Vaclav Kotesovec, Aug 19 2013, updated Oct 11 2018 a(n) = Sum_{i=0..floor(n/2)} C(2*n-4*i+1,i)*C(2*n-4*i+1,n-2*i)/(2*n-4*i+1). - Vladimir Kruchinin, Oct 11 2018
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 24*x^4 + 80*x^5 + 278*x^6 + 997*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 73*x^4 + 256*x^5 + 924*x^6 + 3414*x^7 +...
where A(x) = 1+x^2 + x*(1+x^2)*A(x)^2.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x/A^2)*A*x + (1 + 2^2*x/A^2 + x^2/A^4)*A^2*x^2/2 +
(1 + 3^2*x/A^2 + 3^2*x^2/A^4 + x^3/A^6)*A^3*x^3/3 +
(1 + 4^2*x/A^2 + 6^2*x^2/A^4 + 4^2*x^3/A^6 + x^4/A^8)*A^4*x^4/4 +
(1 + 5^2*x/A^2 + 10^2*x^2/A^4 + 10^2*x^3/A^6 + 5^2*x^4/A^8 + x^5/A^10)*A^5*x^5/5 +
(1 + 6^2*x/A^2 + 15^2*x^2/A^4 + 20^2*x^3/A^6 + 15^2*x^4/A^8 + 6^2*x^5/A^10 + x^6/A^12)*A^6*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 16*x^3/3 + 57*x^4/4 + 231*x^5/5 + 938*x^6/6 + 3830*x^7/7 + 15833*x^8/8 +...
MATHEMATICA
nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x^2)*(1+x*AGF^2)-AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Aug 19 2013 *) CoefficientList[Series[(1 - Sqrt[1 - 4*x*(1 + x^2)^2]) / (2*x*(1 + x^2)), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 11 2018 *) Table[Sum[Binomial[2*n - 4*i + 1, i] * Binomial[2*n - 4*i + 1, n - 2*i]/(2*n - 4*i + 1), {i, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 11 2018, after Vladimir Kruchinin *)
PROG
(PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*x*(1+x^2 +x*O(x^n))^2)) / (2*x*(1+x^2 +x*O(x^n))), n)}
for(n=0, 31, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^2)*(1+x^2)+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j/A^(2*j))*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A^2)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^(2*j))*x^m*A^m/m))); polcoeff(A, n, x)}
G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^5).
+0
1
1, 1, 4, 21, 126, 819, 5611, 39900, 291719, 2179181, 16560175, 127617168, 994951887, 7833555324, 62196300997, 497425570173, 4003607595960, 32404662671330, 263586896132154, 2153631763231319, 17666722629907960, 145449082369322208, 1201414340736684702
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2 and q=2.
FORMULA
G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) ))/2 ) ).
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) is the g.f. of A200075. (2) A(x) = sqrt( (1/x)*Series_Reversion( x/G(x)^2 ) ) where A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) and 1+x*G(x) is the g.f. of A004148. (3) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(2*k)] * x^n*A(x)^(2*n)/n ).
(4) A(x) = exp( Sum_{n>=1} [(1-x*A(x)^2)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(2*k) )] * x^n*A(x)^(2*n)/n ).
a(n) ~ s * sqrt((1 + 2*r*s^2 + 3*r^2*s^5) / (Pi*(3 + 10*r*s^2 + 28*r^2*s^5))) / (2*n^(3/2)*r^n), where r = 0.1130413665724951344267888513870607581680912144315... and s = 1.385648922830296011590145919380626723251960276539... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^5) = s, r*s^2*(3 + 5*r*s^2 + 8*r^2*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 819*x^5 + 5611*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 88*x^3 + 564*x^4 + 3828*x^5 + 27040*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 195*x^3 + 1335*x^4 + 9486*x^5 + 69305*x^6 +...
A(x)^8 = 1 + 8*x + 60*x^2 + 448*x^3 + 3374*x^4 + 25704*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^5 + x^3*A(x)^8.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x)^2)*x*A(x)^2 +
(1 + 2^2*x*A(x)^2 + x^2*A(x)^4)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x)^2 + 3^2*x^2*A(x)^4 + x^3*A(x)^6)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x)^2 + 6^2*x^2*A(x)^4 + 4^2*x^3*A(x)^6 + x^4*A(x)^8)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x)^2 + 10^2*x^2*A(x)^4 + 10^2*x^3*A(x)^6 + 5^2*x^4*A(x)^8 + x^5*A(x)^10)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x)^2 + 15^2*x^2*A(x)^4 + 20^2*x^3*A(x)^6 + 15^2*x^4*A(x)^8 + 6^2*x^5*A(x)^10 + x^6*A(x)^12)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 52*x^3/3 + 403*x^4/4 + 3211*x^5/5 + 26050*x^6/6 +...
MATHEMATICA
CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1-2*x-x^2+x^4 + (1-x-x^2) * Sqrt[(1+x+x^2)*(1-3*x+x^2)])/2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)
PROG
(PARI) {a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n)))/2 ) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(p=2, q=2, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^4).
+0
1
1, 1, 4, 20, 114, 703, 4565, 30752, 212921, 1505916, 10833164, 79018804, 583062388, 4344431508, 32641910199, 247033970128, 1881402836376, 14408753414558, 110897147057354, 857307054338476, 6653979156676983, 51831065993122915, 405060413133136902, 3175019470333290488
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2 and q=1.
FORMULA
G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x-x^3)^2/(1+x^2)^2 ) ).
(2) A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
(3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(2*n+2) / (n+1).
(4) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k] * x^n*A(x)^(2*n)/n ).
(5) A(x) = exp( Sum_{n>=1} [(1-x*A(x))^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k )] * x^n*A(x)^(2*n)/n ).
(6) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = (1 + x*G(x)^2)*(1 + x^2*G(x)^2) is the g.f. of A199874. a(n) ~ s * sqrt((1 + 2*r*s + 3*r^2*s^4) / (3*Pi*(1 + 2*r*s + 7*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.1194948955213353102456218138370139612914667337222... and s = 1.428770161302757679335810379290625953730830139744... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^4) = s, r*s^2*(3 + 4*r*s + 7*r^2*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 114*x^4 + 703*x^5 + 4565*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 522*x^4 + 3381*x^5 + 22735*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 132*x^3 + 841*x^4 + 5588*x^5 + 38288*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 343*x^3 + 2429*x^4 + 17430*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^4 + x^3*A(x)^7.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x))*x*A(x)^2 + (1 + 2^2*x*A(x) + x^2*A(x)^2)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x) + 3^2*x^2*A(x)^2 + x^3*A(x)^3)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x) + 6^2*x^2*A(x)^2 + 4^2*x^3*A(x)^3 + x^4*A(x)^4)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x) + 10^2*x^2*A(x)^2 + 10^2*x^3*A(x)^3 + 5^2*x^4*A(x)^4 + x^5*A(x)^5)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x) + 15^2*x^2*A(x)^2 + 20^2*x^3*A(x)^3 + 15^2*x^4*A(x)^4 + 6^2*x^5*A(x)^5 + x^6*A(x)^6)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 49*x^3/3 + 359*x^4/4 + 2706*x^5/5 + 20767*x^6/6 +...
MATHEMATICA
CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1 - x - x^3)^2/(1 + x^2)^2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)
PROG
(PARI) {a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-x-x^3)^2/(1+x^2+x*O(x^n))^2 ) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(p=2, q=1, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^6).
+0
7
1, 1, 4, 22, 139, 953, 6894, 51796, 400269, 3161262, 25403536, 207043048, 1707345547, 14219399626, 119431172630, 1010495472960, 8604568715969, 73683710894255, 634142349130800, 5482062214763436, 47582484748270453, 414503778412715065, 3622792181209018168, 31758958747482608912
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2, q=3.
FORMULA
G.f. A(x) satisfies:
(1) a(n) = [x^n] (1 + x + x^2 + x^3)^(3*n+1) / (3*n+1).
(2) A(x) = ( (1/x)*Series_Reversion( x/(1 + x + x^2 + x^3)^3 ) )^(1/3).
(3) A( x/(1 + x + x^2 + x^3)^3 ) = 1 + x + x^2 + x^3.
(4) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) = g.f. of A036765 (number of rooted trees with a degree constraint). (5) A(x) = exp( Sum_{n>=1} x^n*A(x)^(2*n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(3*k)] ).
(6) A(x) = exp( Sum_{n>=1} x^n*A(x)^(2*n)/n * [(1-x*A(x)^2)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(3*k) )] ).
a(n) = 1/(3*n + 1)*Sum_{k = 0..floor(n/2)} binomial(3*n + 1,k)*binomial(3*n + 1,n - 2*k).
More generally, the coefficient of x^n in A(x)^r equals r/(3*n + r)*Sum_{k = 0..floor(n/2)} binomial(3*n + r,k)*binomial(3*n + r,n - 2*k) by the Lagrange-Bürmann formula.
O.g.f. A(x) = exp(Sum_{n >= 1} 1/3*b(n)x^n/n), where b(n) = Sum_{k = 0..floor(n/2)} binomial(3*n,k)*binomial(3*n,n - 2*k). Cf. A036765, A186241, A198951. (End) Recurrence: 128*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*(8*n - 3)*(8*n - 1)*(8*n + 1)*(8*n + 3)*(511073753*n^7 - 4871850365*n^6 + 19478089219*n^5 - 42349790393*n^4 + 54094962928*n^3 - 40605677522*n^2 + 16589611340*n - 2846611200)*a(n) = 3*(3*n - 2)*(3*n - 1)*(3047149994898003*n^13 - 32094344705469618*n^12 + 145743661212727337*n^11 - 373710048777443810*n^10 + 593788894662012231*n^9 - 600683242386376410*n^8 + 377600776651518819*n^7 - 130595257353511374*n^6 + 11334217618972546*n^5 + 8004135084547148*n^4 - 2618300200112616*n^3 + 152383960257264*n^2 + 33025238671680*n - 3264156403200)*a(n-1) - 576*(n-1)*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(495741540410*n^10 - 3982082543435*n^9 + 12891395244590*n^8 - 21360691645174*n^7 + 18695904340190*n^6 - 7495052530111*n^5 + 212344193250*n^4 + 656210670544*n^3 - 106487698440*n^2 - 7969373424*n + 1477828800)*a(n-2) + 110592*(n-2)*(n-1)*(3*n - 8)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(511073753*n^7 - 1294334094*n^6 + 979535842*n^5 - 149518418*n^4 - 72732399*n^3 + 16154432*n^2 + 843684*n - 192240)*a(n-3). - Vaclav Kotesovec, Nov 17 2017 a(n) ~ s/(2*sqrt(3*Pi*(4 - 9*r*s^2*(1 + r*s^3)))*n^(3/2)*r^n), where r = 0.1068159753611743655799981945670627355827110854720... and s = 1.345561337338583233012136458010090420775336284226... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^6) = s, 3*r*s^2*(1 + 2*r*s^3 + 3*r^2*s^6) = 1. - Vaclav Kotesovec, Nov 22 2017
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 139*x^4 + 953*x^5 + 6894*x^6 +...
where A(x) = (1 + x*A(x)^3)*(1 + x^2*A(x)^6).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 609*x^4 + 4335*x^5 + 32197*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 272*x^3 + 1989*x^4 + 15054*x^5 + 116955*x^6 +...
A(x)^9 = 1 + 9*x + 72*x^2 + 570*x^3 + 4545*x^4 + 36639*x^5 + 298662*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^3)*x*A^2 + (1 + 2^2*x*A^3 + x^2*A^6)*x^2*A^4/2 +
(1 + 3^2*x*A^3 + 3^2*x^2*A^6 + x^3*A^9)*x^3*A^6/3 +
(1 + 4^2*x*A^3 + 6^2*x^2*A^6 + 4^2*x^3*A^9 + x^4*A^12)*x^4*A^8/4 +
(1 + 5^2*x*A^3 + 10^2*x^2*A^6 + 10^2*x^3*A^9 + 5^2*x^4*A^12 + x^5*A^15)*x^5*A^10/5 + ...
which involves squares of binomial coefficients.
MATHEMATICA
nmax = 23; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x A[x]^3)*(1 + x^2 A[x]^6) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
PROG
(PARI) {a(n)=polcoeff( ((1/x)*serreverse(x/(1 + x + x^2 + x^3 +x*O(x^n))^3))^(1/3), n)}
(PARI) {a(n)=polcoeff( (1 + x + x^2 + x^3 +x*O(x^n))^(3*n+1)/(3*n+1), n)}
(PARI) {a(n)=local(p=2, q=3, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
CROSSREFS
Cf. A036765, A200716, A200717, A200718, A200719, A200725, A200074, A200075, A199874, A199876, A199877, A198951, A198953, A198957, A192415, A198888. Cf. A186241.
G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)).
+0
8
1, 1, 4, 17, 84, 453, 2574, 15185, 92119, 571022, 3600981, 23029021, 149000790, 973581692, 6415198045, 42580369370, 284427460919, 1910594331920, 12898153658337, 87461992473577, 595455441375978, 4068652368270955, 27891991988552554, 191783482751813061, 1322319472577803761
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^(2*k)] ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [(1-x/A(x)^2)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2 * x^k/A(x)^(2*k)] ).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 7.342019160707096169... is the root of the equation -27 + 108*d^2 - 162*d^4 + 54*d^5 + 108*d^6 + 216*d^7 - 27*d^8 - 18*d^9 - 27*d^10 + 4*d^11 = 0 and c = 0.468554406193087607276981923311829947714908080994... - Vaclav Kotesovec, Sep 19 2013
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 84*x^4 + 453*x^5 + 2574*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 76*x^3 + 414*x^4 + 2370*x^5 + 14047*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 120*x^3 + 685*x^4 + 4048*x^5 + 24558*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x) + x^3*A(x)^4.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x/A^2)*x*A^2 + (1 + 2^2*x/A^2 + x^2/A^4)*x^2*A^4/2 +
(1 + 3^2*x/A^2 + 3^2*x^2/A^4 + x^3/A^6)*x^3*A^6/3 +
(1 + 4^2*x/A^2 + 6^2*x^2/A^4 + 4^2*x^3/A^6 + x^4/A^8)*x^4*A^8/4 +
(1 + 5^2*x/A^2 + 10^2*x^2/A^4 + 10^2*x^3/A^6 + 5^2*x^4/A^8 + x^5/A^10)*x^5*A^10/5 + ...
MATHEMATICA
nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF^3) * (1 + x^2*AGF) - AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 19 2013 *)
PROG
(PARI) {a(n)=local(p=2, q=-2, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=-2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=-2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
CROSSREFS
Cf. A200717, A200718, A200719, A200074, A200075, A199874, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^2).
+0
9
1, 1, 4, 18, 93, 521, 3073, 18806, 118297, 760162, 4968480, 32928392, 220766739, 1494635330, 10203884795, 70167751762, 485574854049, 3379064343829, 23631314301088, 165998001901786, 1170706810318259, 8286253163771045, 58842370488310336, 419102145275264242, 2993221125640617827
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ), then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).
FORMULA
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x*(1 + sqrt(1 - 4*x*(1+x^2)^2)) / (2*(1+x^2)) ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^k] ).
(3) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [(1-x/A(x)^2)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k / A(x)^k] ).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 7.60435909657327146... is the root of the equation -108 + 27*d^2 + 1620*d^3 - 216*d^4 - 1456*d^5 - 2556*d^6 - 716*d^7 + 20*d^8 + 16*d^9 = 0 and c = 0.45780648099092640511434469483084555191269495951... - Vaclav Kotesovec, Sep 19 2013
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 93*x^4 + 521*x^5 + 3073*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 238*x^4 + 1372*x^5 + 8256*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 79*x^3 + 447*x^4 + 2655*x^5 + 16324*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 180*x^3 + 1110*x^4 + 7006*x^5 + 45075*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*A(x)^5.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x/A)*x*A^2 + (1 + 2^2*x/A + x^2/A^2)*x^2*A^4/2 +
(1 + 3^2*x/A + 3^2*x^2/A^2 + x^3/A^3)*x^3*A^6/3 +
(1 + 4^2*x/A + 6^2*x^2/A^2 + 4^2*x^3/A^3 + x^4/A^4)*x^4*A^8/4 +
(1 + 5^2*x/A + 10^2*x^2/A^2 + 10^2*x^3/A^3 + 5^2*x^4/A^4 + x^5/A^5)*x^5*A^10/5 + ...
MATHEMATICA
nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF^3) * (1 + x^2*AGF^2) - AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Sep 19 2013 *)
PROG
(PARI) {a(n)=polcoeff((1/x)*serreverse( x*(1 + sqrt(1 - 4*x*(1+x^2)^2 +x*O(x^n))) / (2*(1+x^2)) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(p=2, q=-1, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=-1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=-1, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
CROSSREFS
Cf. A200716, A200718, A200719, A200074, A200075, A215576, A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x^2*A(x)^6).
+0
13
1, 1, 3, 14, 75, 433, 2636, 16668, 108399, 720431, 4871555, 33409042, 231817448, 1624503716, 11480658056, 81731416480, 585579734959, 4219179476875, 30552067317233, 222225174139730, 1622894404239115, 11894991079960721, 87472260252499560, 645183802300787356, 4771926560361458884
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).
FORMULA
G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( 2*x^5*(1+x)^2/(1 - 2*x^2*(1+x)^2 - sqrt(1 - 4*x^2*(1+x)^2)) ) ).
(2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A104545 (Motzkin paths of length n having no consecutive (1,0) steps). (3) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(4*k)] ).
(4) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [(1-x/A(x)^4)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k * A(x)^(4*k)] ).
a(n) = Sum_{k=0..floor(n/2)}((binomial(2*n+2*k+1,k)*binomial(2*n+2*k+1,n-2*k))/(2*n+2*k+1)). - Vladimir Kruchinin, Mar 11 2016
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 75*x^4 + 433*x^5 + 2636*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 187*x^4 + 1100*x^5 + 6784*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 194*x^3 + 1200*x^4 + 7674*x^5 + 50317*x^6 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 336*x^3 + 2210*x^4 + 14776*x^5 + 100216*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^6 + x^3*A(x)^8.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^4)*x*A + (1 + 2^2*x*A^4 + x^2*A^8)*x^2*A^2/2 +
(1 + 3^2*x*A^4 + 3^2*x^2*A^8 + x^3*A^12)*x^3*A^3/3 +
(1 + 4^2*x*A^4 + 6^2*x^2*A^8 + 4^2*x^3*A^12 + x^4*A^16)*x^4*A^4/4 +
(1 + 5^2*x*A^4 + 10^2*x^2*A^8 + 10^2*x^3*A^12 + 5^2*x^4*A^16 + x^5*A^20)*x^5*A^5/5 + ...
The g.f. of A104545, G(x) = A(x/G(x)^2) where A(x) = G(x*A(x)^2), begins: G(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 55*x^7 + 129*x^8 +...
MATHEMATICA
a[n_] := Sum[Binomial[2*n + 2*k + 1, k]*Binomial[2*n + 2*k + 1, n - 2*k]/ (2*n + 2*k + 1), {k, 0, n/2}];
PROG
(PARI) {a(n)=polcoeff(sqrt( (1/x)*serreverse( 2*x^5*(1+x)^2/(1 - 2*x^2*(1+x)^2 - sqrt(1 - 4*x^2*(1+x)^2+O(x^(n+6)))) ) ), n)}
(PARI) {a(n)=local(p=1, q=4, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=1, q=4, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=1, q=4, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(Maxima)
a(n):=sum((binomial(2*n+2*k+1, k)*binomial(2*n+2*k+1, n-2*k))/(2*n+2*k+1), k, 0, (n)/2); /* Vladimir Kruchinin, Mar 11 2016 */
CROSSREFS
Cf. A104545, A200716, A200717, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x^2*A(x)^5).
+0
12
1, 1, 3, 13, 64, 340, 1903, 11053, 65993, 402527, 2497439, 15712220, 100001459, 642719263, 4165537744, 27193644061, 178654643151, 1180282875483, 7836312619243, 52259258911091, 349902441457427, 2351240866736891, 15851508780927739, 107187240225220684, 726784821098903319
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(3*k)] ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [(1-x/A(x)^3)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k * A(x)^(3*k)] ).
Recurrence: 4232*(n-2)*(n-1)*n*(2*n - 3)*(2*n - 1)*(2*n + 1)*(108983978975*n^7 - 1828734495225*n^6 + 13017379495661*n^5 - 50928975062019*n^4 + 118201965098732*n^3 - 162617590602876*n^2 + 122676758610192*n - 39103265134080)*a(n) = 8*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(850837923857825*n^9 - 15127768128079400*n^8 + 116088908648008427*n^7 - 502364025369222635*n^6 + 1342887860190877280*n^5 - 2280899268898038065*n^4 + 2433907848768834828*n^3 - 1548429898790214180*n^2 + 521138603722292640*n - 68863424146977600)*a(n-1) - 30*(n-2)*(2*n - 3)*(155302170039375*n^11 - 3227155335853125*n^10 + 29807524885054600*n^9 - 161278340404759950*n^8 + 566950865855228019*n^7 - 1356848300481904461*n^6 + 2250482361655315470*n^5 - 2579665279074165840*n^4 + 1996011605601581864*n^3 - 988803599084885136*n^2 + 280851990522009984*n - 34444332223983360)*a(n-2) + 5*(7288303593953125*n^13 - 187891351713750000*n^12 + 2204843674914291875*n^11 - 15579013461781304250*n^10 + 73867718896175411475*n^9 - 247858726321141236540*n^8 + 604530296941440837821*n^7 - 1082990060568950070282*n^6 + 1421457900098213642392*n^5 - 1345695224728829837040*n^4 + 889319601933492222864*n^3 - 386196670582228097568*n^2 + 97916706472751405568*n - 10797892365692920320)*a(n-3) + 10*(n-3)*(2*n - 7)*(5*n - 18)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(108983978975*n^7 - 1065846642400*n^6 + 4333636082786*n^5 - 9458655747964*n^4 + 11899609166891*n^3 - 8554104592084*n^2 + 3208950812340*n - 473478110640)*a(n-4). - Vaclav Kotesovec, Nov 17 2017 a(n) ~ s * sqrt((r*s*(r*s^3 - 1) - 3) / (7*Pi*(5*r*s*(1 + r*s^3) - 3))) / (2*n^(3/2)*r^n), where r = 0.1385102270697349252376651829944449360743895474888... and s = 1.450646440303399446510765649245639306003224666768... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^5) = s, r*s*(2 + 5*r*s^3 + 7*r^2*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n+k+1,n-2*k) / (2*n+k+1). - Seiichi Manyama, Jul 18 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 64*x^4 + 340*x^5 + 1903*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 163*x^4 + 886*x^5 + 5039*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 765*x^4 + 4481*x^5 + 26920*x^6 +...
A(x)^7 = 1 + 7*x + 42*x^2 + 252*x^3 + 1533*x^4 + 9457*x^5 + 59101*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^5 + x^3*A(x)^7.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^3)*x*A + (1 + 2^2*x*A^3 + x^2*A^6)*x^2*A^2/2 +
(1 + 3^2*x*A^3 + 3^2*x^2*A^6 + x^3*A^9)*x^3*A^3/3 +
(1 + 4^2*x*A^3 + 6^2*x^2*A^6 + 4^2*x^3*A^9 + x^4*A^12)*x^4*A^4/4 +
(1 + 5^2*x*A^3 + 10^2*x^2*A^6 + 10^2*x^3*A^9 + 5^2*x^4*A^12 + x^5*A^15)*x^5*A^5/5 + ...
PROG
(PARI) {a(n)=local(p=1, q=3, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=1, q=3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=1, q=3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
CROSSREFS
Cf. A200716, A200717, A200718, A200074, A200075, A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A036765, A104545.
G.f. satisfies: A(x) = (1+x^2)*(1 + x*A(x)^3).
+0
1
1, 1, 4, 16, 76, 399, 2206, 12664, 74790, 451420, 2772313, 17267652, 108821293, 692609446, 4445642625, 28744599748, 187047449289, 1224027357216, 8050074481917, 53179900898596, 352726704965748, 2348036826102013, 15682048658695168, 105052549830928908, 705678173069959645
COMMENTS
More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)); here p=2, q=-3.
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k / A(x)^(3*k)] ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(2*n)/n * [(1 - x/A(x)^3)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k / A(x)^(3*k) )].
Recurrence: 2*(n-4)*(n-2)*n*(2*n+1)*a(n) = 3*(n-4)*(n-2)*(3*n-2)*(3*n-1)*a(n-1) - 2*(n-4)*(n-2)*n*(2*n-1)*a(n-2) + 6*(n-4)*(3*n-8)*(6*n^2 - 17*n + 2)*a(n-3) + 6*(3*n-14)*(9*n^3 - 66*n^2 + 114*n - 4)*a(n-5) + 6*n*(3*n-20)*(6*n^2 - 47*n + 78)*a(n-7) + 3*(n-2)*n*(3*n-26)*(3*n-19)*a(n-9). - Vaclav Kotesovec, Aug 19 2013 a(n) ~ c*d^n/n^(3/2), where d = 7.1535029565... is the root of the equation -27 - 81*d^2 - 81*d^4 - 27*d^6 + 4*d^7 = 0 and c = 0.26300783791885411389369671... - Vaclav Kotesovec, Aug 19 2013
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 76*x^4 + 399*x^5 + 2206*x^6 +...
Related expansion:
A(x)^3 = 1 + 3*x + 15*x^2 + 73*x^3 + 384*x^4 + 2133*x^5 + 12280*x^6 +...
where a(3) = 1 + 15; a(4) = 3 + 73; a(5) = 15 + 384; a(6) = 73 + 2133; ...
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x/A^3)*x*A^2 + (1 + 2^2*x/A^3 + x^2/A^6)*x^2*A^4/2 +
(1 + 3^2*x/A^3 + 3^2*x^2/A^6 + x^3/A^9)*x^3*A^6/3 +
(1 + 4^2*x/A^3 + 6^2*x^2/A^6 + 4^2*x^3/A^9 + x^4/A^12)*x^4*A^8/4 +
(1 + 5^2*x/A^3 + 10^2*x^2/A^6 + 10^2*x^3/A^9 + 5^2*x^4/A^12 + x^5/A^15)*x^5*A^10/5 + ...
which involves the squares of the binomial coefficients C(n,k).
MATHEMATICA
nmax=20; aa=ConstantArray[0, nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n, {n, 1, j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x^2)*(1+x*AGF^3)-AGF, x, j]==0, koef][[1]]; aa[[j]]=koef/.sol[[1]], {j, 2, nmax}]; Flatten[{1, aa}] (* Vaclav Kotesovec, Aug 19 2013 *)
PROG
(PARI) {a(n)=local(p=2, q=-3, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(p=2, q=-3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
(PARI) {a(n)=local(p=2, q=-3, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}
CROSSREFS
Cf. A200716, A200717, A200718, A200719, A200074, A200075, A199874, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.
G.f. satisfies A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)).
+0
12
1, 1, 3, 9, 30, 108, 406, 1577, 6280, 25499, 105169, 439388, 1855636, 7908909, 33975250, 146954693, 639460707, 2797384235, 12295494109, 54272825103, 240480529815, 1069257987503, 4769306203838, 21334400243252, 95687482105807, 430217846136134, 1938651904470374, 8754225470415889
COMMENTS
More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).
FORMULA
G.f. satisfies:
(1) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(n-k)] * x^n/n ).
(2) A(x) = exp( Sum_{n>=1} [(1-x/A(x))^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k/A(x)^k )] * x^n*A(x)^n/n.
(3) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A199876. (4) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A199876. Recurrence: (n+1)*(n+2)*(1241*n^4 - 10636*n^3 + 25417*n^2 - 7382*n - 17136)*a(n) = - 18*(n+1)*(443*n^3 - 3889*n^2 + 9734*n - 5712)*a(n-1) + 4*(6205*n^6 - 53180*n^5 + 115741*n^4 + 64762*n^3 - 370103*n^2 + 246727*n - 25704)*a(n-2) + 6*(2482*n^6 - 24995*n^5 + 76519*n^4 - 36347*n^3 - 185471*n^2 + 293092*n - 140400)*a(n-3) + 2*(4964*n^6 - 57436*n^5 + 228617*n^4 - 276802*n^3 - 361447*n^2 + 956696*n - 320496)*a(n-4) - 6*(2482*n^6 - 32441*n^5 + 140587*n^4 - 173153*n^3 - 266705*n^2 + 677518*n - 291840)*a(n-5) + 12*(n-4)*(2*n - 11)*(11*n^2 + 73*n - 748)*a(n-6) + 2*(n-5)*(2*n - 13)*(1241*n^4 - 5672*n^3 + 955*n^2 + 16508*n - 8496)*a(n-7). - Vaclav Kotesovec, Aug 18 2013 a(n) ~ c*d^n/n^(3/2), where d = 4.770539985405... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.612892860188927397373456... - Vaclav Kotesovec, Aug 18 2013 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k+1,k) * binomial(2*n-3*k+1,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Jul 18 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 108*x^5 + 406*x^6 + 1577*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 87*x^4 + 330*x^5 + 1289*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 180*x^4 + 720*x^5 + 2928*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^3.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^2*x*A + x^2)*x^2/2 +
(A^3 + 3^2*x*A^2 + 3^2*x^2*A + x^3)*x^3/3 +
(A^4 + 4^2*x*A^3 + 6^2*x^2*A^2 + 4^2*x^3*A + x^4)*x^4/4 +
(A^5 + 5^2*x*A^4 + 10^2*x^2*A^3 + 10^2*x^3*A^2 + 5^2*x^4*A + x^5)*x^5/5 +
(A^6 + 6^2*x*A^5 + 15^2*x^2*A^4 + 20^2*x^3*A^3 + 15^2*x^4*A^2 + 6^2*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 77*x^4/4 + 331*x^5/5 + 1445*x^6/6 + 6392*x^7/7 + 28565*x^8/8 +...
MAPLE
a:= n-> coeff(series(RootOf(A=(1+x*A^2)*(1+x^2*A), A), x, n+1), x, n):
MATHEMATICA
m = 28; A[_] = 0;
Do[A[x_] = (1 + x A[x]^2)(1 + x^2 A[x]) + O[x]^m, {m}];
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^2)*(1+x^2*A^1)+x*O(x^n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^j)*x^m*A^m/m))); polcoeff(A, n, x)}
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