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%I #7 Jul 20 2019 16:33:15
%S 1,5,69,1496,45771,1840537,92925982,5705543791,416015394341,
%T 35365673566750,3454046493504337,382930667897753421,
%U 47708365129614794580,6622948820406278058625,1016977626656613380728781,171637260767262574245781800,31661205827344145981298200207,6352045190999137085697971335893
%N E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
%C More generally, the following sums are equal:
%C (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
%C (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
%C here, q = exp(x), p = 3, r = 1.
%F E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
%F E.g.f.: exp(-4) * Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x) ) / n!.
%F FORMULAS FOR TERMS.
%F a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.
%e E.g.f.: A(x) = 1 + 5*x + 69*x^2/2! + 1496*x^3/3! + 45771*x^4/4! + 1840537*x^5/5! + 92925982*x^6/6! + 5705543791*x^7/7! + 416015394341*x^8/8! + 35365673566750*x^9/9! + 3454046493504337*x^10/10! + ...
%e such that
%e A(x) = exp(-4) * (1 + (exp(x) + 3) + (exp(2*x) + 3)^2/2! + (exp(3*x) + 3)^3/3! + (exp(4*x) + 3)^4/4! + (exp(5*x) + 3)^5/5! + (exp(6*x) + 3)^6/6! + ...)
%e also
%e A(x) = exp(-4) * (exp(3) + exp(x)*exp(3*exp(x)) + exp(4*x)*exp(3*exp(2*x))/2! + exp(9*x)*exp(3*exp(3*x))/3! + exp(16*x)*exp(3*exp(4*x))/4! + exp(25*x)*exp(3*exp(5*x))/5! + exp(36*x)*exp(3*exp(6*x))/6! + ...).
%o (PARI) /* Requires suitable precision */
%o \p200
%o Vec(round(serlaplace( exp(-4) * sum(n=0, 500, (exp(n*x +O(x^31)) + 3)^n/n! ) )))
%Y Cf. A326600, A020557, A326433, A326434, A326436, A326437.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 11 2019
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