A266524 - OEIS

%I #8 Dec 31 2015 00:12:23

%S 1,4,100,4464,286816,24053120,2488967136,306383969920,43726697867008,

%T 7098711727021056,1291743506952832000,260410631081389420544,

%U 57609344863582419640320,13875489289115958335143936,3614364399291754755286614016,1012444950785630853817442304000,303479487751656117544078504493056,96925825525767333731669511270563840,32859305845564004294368688506268024832,11784943093649049136596829229809817092096,4458038385946160559288726139220234076160000,1773928724624151210275576625473634276174987264,740706616375525604793089813921394696991733186560

%N E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).

%C The e.g.f. A(x) of this sequence also satisfies:

%C A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(4*n) * (x/N^3)^n/n! ] / G(x,y)^N

%C where

%C G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(4*n) * (x/N^3)^n/n! ]^(1/N)

%C for all real y.

%e E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 4464*x^3/3! + 286816*x^4/4! + 24053120*x^5/5! + 2488967136*x^6/6! + 306383969920*x^7/7! + 43726697867008*x^8/8! + 7098711727021056*x^9/9! + 1291743506952832000*x^10/10! +...

%e such that

%e A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ] / F(x)^N

%e where

%e F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N)

%e and

%e F(x) =  1 + x + 9*x^2/2! + 205*x^3/3! + 8033*x^4/4! + 456561*x^5/5! + 34307545*x^6/6! + 3219222301*x^7/7! + 363018204225*x^8/8! + 47866764942721*x^9/9! + 7230829461286121*x^10/10! +...+ A266483(n)*x^n/n! +...

%Y Cf. A266483, A266522, A266523, A266525.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 30 2015