%I #30 Nov 30 2016 20:35:01

%S 1,1,2,7,26,102,424,1827,8078,36466,167376,778718,3664164,17407068,

%T 83375616,402198915,1952296598,9528757098,46735576816,230227356906,

%U 1138609205372,5651170500612,28138939936704,140527262919342,703704207921932,3532664478314484,17775185122527776

%N G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^2*A(x)^4).

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

%F (1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(3*k) ).

%F (2) A(x) = (1/x)*Series_Reversion( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x)^2)) ).

%F (3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A104545 (Motzkin paths of length n having no consecutive (1,0) steps).

%F (4) A(x) = exp( Sum_{n>=1} 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) ).

%F a(n) = sum(j=0..n/2, binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1)). - _Vladimir Kruchinin_, May 28 2014

%F a(n) ~ sqrt((1 + 2*r*s^3 + 3*r^2*s^4)/(2*Pi*s*(3 + 5*r*s))) / (2*n^(3/2)*r^(n+1/2)), where r = 0.187614989725738719..., s = 1.61178302212918247... are roots of the system of equations r + 4*r^2*s^3 + 5*r^3*s^4 = 1, (1+r*s)*(1+r^2*s^4) = s. - _Vaclav Kotesovec_, May 28 2014

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 102*x^5 + 424*x^6 + 1827*x^7 +...

%e Related expansions:

%e A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1028*x^5 + 4570*x^6 +...

%e A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 375*x^4 + 1681*x^5 + 7660*x^6 +...

%e where A(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^5.

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

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

%e (1 + 3^2*x*A(x)^3 + 3^2*x^2*A(x)^6 + x^3*A(x)^9)*x^3/3 +

%e (1 + 4^2*x*A(x)^3 + 6^2*x^2*A(x)^6 + 4^2*x^3*A(x)^9 + x^4*A(x)^12)*x^4/4 +

%e (1 + 5^2*x*A(x)^3 + 10^2*x^2*A(x)^6 + 10^2*x^3*A(x)^9 + 5^2*x^4*A(x)^12 + x^5*A(x)^15)*x^5/5 +...

%e more explicitly,

%e log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 75*x^4/4 + 356*x^5/5 + 1746*x^6/6 + 8660*x^7/7 + 43299*x^8/8 +...

%e Also, g.f. A(x) = G(x*A(x)) where G(x) = A(x/G(x)) (g.f. of A104545) begins:

%e G(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 55*x^7 + 129*x^8 +...

%t CoefficientList[1/x*InverseSeries[Series[2*x^3*(1+x)/(1 - Sqrt[1-4*x^2*(1+x)^2]), {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, May 28 2014 *)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^2*(A+x*O(x^n))^4)); polcoeff(A, n)}

%o (PARI) {a(n)=polcoeff((1/x)*serreverse( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x +x^3*O(x^n))^2))), n)}

%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*(A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x*A^3)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

%o (Maxima) a(n):=sum(binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1),j,0,(n)/2); /* _Vladimir Kruchinin_, May 28 2014 */

%o (PARI)

%o x='x; y='y; Fxy = (1 + x*y)*(1 + x^2*y^4) - y;

%o seq(N) = {

%o my(y0 = 1 + O('x^N), y1=0);

%o for (k = 1, N,

%o y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);

%o if (y1 == y0, break()); y0 = y1);

%o Vec(y0);

%o };

%o seq(27) \\ _Gheorghe Coserea_, Nov 30 2016

%Y Cf. A198953, A198951, A007863, A036765, A104545.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 01 2011