%I #30 Jul 18 2023 10:42:14

%S 1,1,3,10,37,147,611,2625,11564,51953,237123,1096420,5125063,24178427,

%T 114974387,550511901,2651896733,12843003108,62494595022,305400429548,

%U 1498184696271,7375179807191,36421312544431,180383163330765,895756907248150,4459095182031675,22247684478181317

%N G.f. satisfies A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)^2).

%H Seiichi Manyama, <a href="/A199874/b199874.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = (1/x)*Series_Reversion( x/(1+x^2) - x^2 ).

%F (2) A( x*(1-x-x^3)/(1+x^2) ) = (1+x^2)/(1-x-x^3).

%F (3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(n+1) / (n+1).

%F (4) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k] * x^n*A(x)^n/n ).

%F (5) A(x) = exp( Sum_{n>=1} [(1-x)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k )] * x^n*A(x)^n/n ).

%F Recurrence: 31*(n-1)*n*(n+1)*(85396*n^4 - 902916*n^3 + 3471647*n^2 - 5767203*n + 3503250)*a(n) = 2*(n-1)*n*(6319304*n^5 - 69975436*n^4 + 290875210*n^3 - 559740413*n^2 + 484175751*n - 138985722)*a(n-1) + 2*(n-1)*(2903464*n^6 - 36506072*n^5 + 179801738*n^4 - 439606930*n^3 + 553204983*n^2 - 328951215*n + 67014378)*a(n-2) + 2*(2*n - 5)*(1964108*n^6 - 24695284*n^5 + 123902749*n^4 - 317652203*n^3 + 438313617*n^2 - 307740825*n + 85471038)*a(n-3) - 32*(n-3)*(2*n - 7)*(85396*n^5 - 860218*n^4 + 3249611*n^3 - 5747414*n^2 + 4753791*n - 1471338)*a(n-4) + 8*(n-4)*(n-3)*(2*n - 9)*(85396*n^4 - 561332*n^3 + 1275275*n^2 - 1191073*n + 390174)*a(n-5). - _Vaclav Kotesovec_, Aug 18 2013

%F a(n) ~ c*d^n/n^(3/2), where d=5.28245622984... is the root of the equation -16 + 64*d - 92*d^2 - 68*d^3 - 148*d^4 + 31*d^5 = 0 and c = 0.49559010377906722118329... - _Vaclav Kotesovec_, Aug 18 2013

%F 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). - _Seiichi Manyama_, Jul 18 2023

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 147*x^5 + 611*x^6 +...

%e where A( x/(1+x^2) - x^2 ) = (1+x^2)/(1-x-x^3).

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 26*x^3 + 103*x^4 + 428*x^5 + 1838*x^6 +...

%e A(x)^4 = 1 + 4*x + 18*x^2 + 80*x^3 + 359*x^4 + 1632*x^5 + 7506*x^6 +...

%e where A(x) = 1 + x*(1+x)*A(x)^2 + x^3*A(x)^4.

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + x)*x*A(x) + (1 + 2^2*x + x^2)*x^2*A(x)^2/2 +

%e (1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^3/3 +

%e (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^4/4 +

%e (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^5/5 +

%e (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^6*A(x)^6/6 +...

%e more explicitly,

%e log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 481*x^5/5 + 2330*x^6/6 +...

%t 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^2)*(1+x^2*AGF^2)-AGF,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* _Vaclav Kotesovec_, Aug 18 2013 *)

%o (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)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

%o (PARI) {a(n)=polcoeff((1/x)*serreverse(x/(1+x^2+x*O(x^n))-x^2),n)}

%o (PARI) {a(n)=polcoeff(((1+x^2)/(1-x-x^3+x*O(x^n)))^(n+1)/(n+1), n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j)*x^m*A^m/m))); polcoeff(A, n, x)}

%Y Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

%Y Cf. A186241, A200074, A200075, A200718, A200719, A215576.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 11 2011