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%I #11 Dec 09 2014 00:38:10
%S 1,3,26,430,10872,373664,16295152,862486944,53729041280,3851892172288,
%T 312411790027776,28284076403710208,2827642792215049216,
%U 309396856974126428160,36777992050266076762112,4719560392385576181243904,650284066459536965937364992,95752333835299098922624876544,15005473998204120386383308390400
%N E.g.f.: exp(4*x*G(x)^3) / G(x) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A'(x)/A(x) = G(x)^3 + 2*G'(x)/G(x).
%F (2) A(x) = F(x/A(x)^3) where F(x) is the e.g.f. of A251694.
%F (3) A(x) = Sum_{n>=0} A251694(n)*(x/A(x)^3)^n/n! where A251694(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.
%F (4) [x^n/n!] A(x)^(3*n+1) = (2*n+1) * (3*n+1)^(n-1) * 4^n.
%F a(n) = Sum_{k=0..n} 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) for n>=0.
%F Recurrence: 3*(3*n-2)*(3*n-1)*(64*n^3 - 344*n^2 + 598*n - 315)*a(n) = 8*(2048*n^6 - 16128*n^5 + 51136*n^4 - 82160*n^3 + 67332*n^2 - 22212*n - 855)*a(n-1) + 256*(64*n^3 - 152*n^2 + 102*n + 3)*a(n-2). - _Vaclav Kotesovec_, Dec 07 2014
%F a(n) ~ 2^(8*n-2) / 3^(3*n-1/2) * n^(n-1) / exp(n-1). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
%e such that A(x) = exp(4*x*G(x)^3) / G(x)
%e where G(x) = 1 + x*G(x)^4 is the g.f. of A002293:
%e G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%t Table[Sum[4^k * n!/k! * Binomial[4*n-k-2,n-k] * (3*k-1)/(3*n-1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 07 2014 *)
%o (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^4 +x*O(x^n)); n!*polcoeff(exp(4*x*G^3)/G, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0, n, 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251574, A251694, A002293.
%Y Cf. Variants: A243953, A251663, A251665, A251666, A251667, A251668, A251669, A251670.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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