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%I #14 Aug 18 2013 13:00:53
%S 1,1,4,19,103,604,3728,23866,157015,1055121,7211227,49970893,
%T 350283538,2479386646,17696379769,127220129807,920376055978,
%U 6695598216385,48950677790253,359456740745360,2650089982411601,19608110285151138,145555836451997536,1083723594296492362
%N G.f. satisfies: A(x) = (1 + x*A(x)^3)*(1 + x^2*A(x)^3).
%H Vaclav Kotesovec, <a href="/A199876/b199876.txt">Table of n, a(n) for n = 0..300</a>
%H Vaclav Kotesovec, <a href="/A199876/a199876.txt">Recurrence</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k] * x^n*A(x)^(2*n)/n ).
%F (2) 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)^(2*n)/n.
%F a(n) ~ c*d^n/n^(3/2), where d=7.9486365297943819... is the root of the equation -729 + 4374*d - 10827*d^2 + 13770*d^3 + 13095*d^4 + 28404*d^5 - 4664*d^6 + 108*d^7 = 0 and c = 0.2415824543... (note that the term with root d=35.7258 tends to zero). - _Vaclav Kotesovec_, Aug 18 2013
%e G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 103*x^4 + 604*x^5 + 3728*x^6 + 23866*x^7 +...
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 15*x^2 + 82*x^3 + 483*x^4 + 2991*x^5 + 19192*x^6 +...
%e A(x)^6 = 1 + 6*x + 39*x^2 + 254*x^3 + 1683*x^4 + 11340*x^5 + 77544*x^6 +...
%e where A(x) = 1 + x*(1+x)*A(x)^3 + x^3*A(x)^6.
%e The logarithm of the g.f. equals the series:
%e log(A(x)) = (1 + x)*x*A(x)^2 + (1 + 2^2*x + x^2)*x^2*A(x)^4/2 +
%e (1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^6/3 +
%e (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^8/4 +
%e (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^10/5 +
%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)^12/6 +...
%e more explicitly,
%e log(A(x)) = x + 7*x^2/2 + 46*x^3/3 + 319*x^4/4 + 2281*x^5/5 + 16612*x^6/6 + 122571*x^7/7 +...
%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^3)*(1+x^2*AGF^3)-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=(1 + x*A^3)*(1 + x^2*A^3)+x*O(x^n));polcoeff(A,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)*(x*A^2+x*O(x^n))^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)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j)*x^m*A^(2*m)/m))); polcoeff(A, n, x)}
%Y Cf. A199874, A199877.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 11 2011
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