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%I #9 Dec 09 2014 00:39:38
%S 1,5,74,2028,83352,4607496,321156000,27064420704,2677510124928,
%T 304299947999232,39075730095810816,5595805388119057920,
%U 884245579070535235584,152843879008651568329728,28688663318934190485491712,5811091829207760774331662336,1263471121829937070180445552640
%N E.g.f.: exp(6*x*G(x)^5) / G(x) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%F Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A'(x)/A(x) = G(x)^5 + 4*G'(x)/G(x).
%F (2) A(x) = F(x/A(x)^5) where F(x) is the e.g.f. of A251696.
%F (3) A(x) = Sum_{n>=0} A251696(n)*(x/A(x)^5)^n/n! where A251696(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n .
%F (4) [x^n/n!] A(x)^(5*n+1) = (4*n+1) * (5*n+1)^(n-1) * 6^n .
%F a(n) = Sum_{k=0..n} 6^k * n!/k! * binomial(6*n-k-2,n-k) * (5*k-1)/(5*n-1) for n>=0.
%F Recurrence: 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*(1296*n^5 - 11394*n^4 + 40230*n^3 - 71274*n^2 + 63110*n - 21963)*a(n) = 144*(419904*n^10 - 5161320*n^9 + 28223964*n^8 - 90513612*n^7 + 188713962*n^6 - 267339204*n^5 + 259905051*n^4 - 169257762*n^3 + 67929146*n^2 - 12957136*n - 43050)*a(n-1) + 46656*(1296*n^5 - 4914*n^4 + 7614*n^3 - 5988*n^2 + 2156*n + 5)*a(n-2). - _Vaclav Kotesovec_, Dec 07 2014
%F a(n) ~ 4 * 6^(6*n-3/2) / 5^(5*n-1/2) * n^(n-1) / exp(n-1). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
%e such that A(x) = exp(6*x*G(x)^5) / G(x)
%e where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
%e G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
%t Table[Sum[6^k * n!/k! * Binomial[6*n-k-2,n-k] * (5*k-1)/(5*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^6 +x*O(x^n)); n!*polcoeff(exp(6*x*G^5)/G, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0, n, 6^k * n!/k! * binomial(6*n-k-2,n-k) * (5*k-1)/(5*n-1) )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251576, A251696, A002295.
%Y Cf. Variants: A243953, A251663, A251664, A251665, A251667, A251668, A251669, A251670.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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