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A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

%I A326435 #7 Jul 20 2019 16:33:15
%S A326435 1,5,69,1496,45771,1840537,92925982,5705543791,416015394341,
%T A326435 35365673566750,3454046493504337,382930667897753421,
%U A326435 47708365129614794580,6622948820406278058625,1016977626656613380728781,171637260767262574245781800,31661205827344145981298200207,6352045190999137085697971335893
%N A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
%C A326435 More generally, the following sums are equal:
%C A326435 (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
%C A326435 (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
%C A326435 here, q = exp(x), p = 3, r = 1.
%F A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
%F A326435 E.g.f.: exp(-4) * Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x) ) / n!.
%F A326435 FORMULAS FOR TERMS.
%F A326435 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 A326435 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 A326435 such that
%e A326435 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 A326435 also
%e A326435 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 A326435 (PARI) /* Requires suitable precision */
%o A326435 \p200
%o A326435 Vec(round(serlaplace( exp(-4) * sum(n=0, 500, (exp(n*x +O(x^31)) + 3)^n/n! ) )))
%Y A326435 Cf. A326600, A020557, A326433, A326434, A326436, A326437.
%K A326435 nonn
%O A326435 0,2
%A A326435 _Paul D. Hanna_, Jul 11 2019