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%I #5 Jul 20 2019 16:34:27
%S 1,12,298,11154,568004,37059182,2978383982,286712714932,
%T 32370944416718,4216616929161674,625354679867770896,
%U 104450484419292872298,19469192354728354857686,4018460441266469063161936,912287005016859245973405858,226476227666270561445555706042,61164205107875867322971316940164
%N E.g.f.: exp(-5) * Sum_{n>=0} (2*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 = 3/2, r = 2.
%F E.g.f.: exp(-5) * Sum_{n>=0} (2*exp(n*x) + 3)^n / n!.
%F E.g.f.: exp(-5) * Sum_{n>=0} 2^n * exp(n^2*x) * exp( 3*exp(n*x) ) / n!.
%e E.g.f.: A(x) = 1 + 12*x + 298*x^2/2! + 11154*x^3/3! + 568004*x^4/4! + 37059182*x^5/5! + 2978383982*x^6/6! + 286712714932*x^7/7! + 32370944416718*x^8/8! + 4216616929161674*x^9/9! + ...
%e such that
%e A(x) = exp(-5) * (1 + (2*exp(x) + 3) + (2*exp(2*x) + 3)^2/2! + (2*exp(3*x) + 3)^3/3! + (2*exp(4*x) + 3)^4/4! + (2*exp(5*x) + 3)^5/5! + (2*exp(6*x) + 3)^6/6! + ...)
%e also
%e A(x) = exp(-5) * (exp(3) + 2*exp(x)*exp(3*exp(x)) + 2^2*exp(4*x)*exp(3*exp(2*x))/2! + 2^3*exp(9*x)*exp(3*exp(3*x))/3! + 2^4*exp(16*x)*exp(3*exp(4*x))/4! + 2^5*exp(25*x)*exp(3*exp(5*x))/5! + 2^6*exp(36*x)*exp(3*exp(6*x))/6! + ...).
%o (PARI) /* Requires suitable precision */
%o \p200
%o Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (2*exp(n*x +O(x^31)) + 3)^n/n! ) )))
%Y Cf. A326600, A020557, A326433, A326434, A326435, A326436.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 11 2019
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