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A326009 E.g.f.: Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.

%I A326009 #6 Jul 13 2019 12:20:28
%S A326009 1,2,8,56,564,7452,124126,2527646,61337576,1740438008,56893173354,
%T A326009 2116141180650,88637462278492,4144712080864292,214742915441526686,
%U A326009 12247719772739219558,764573919234220965072,51977513845734053953776,3830761480589037404767954,304839727443701572462549058,26096983659506717348854764356,2395544800795092178844224643612,235073598248121646307555752669446
%N A326009 E.g.f.: Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.
%C A326009 More generally, the following sums are equal:
%C A326009 (1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
%C A326009 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326009 here, q = exp(x), p = exp(-x), r = exp(x)*x.
%F A326009 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals the following sums.
%F A326009 (1) Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!,
%F A326009 (2) Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n / n!.
%e A326009 E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 564*x^4/4! + 7452*x^5/5! + 124126*x^6/6! + 2527646*x^7/7! + 61337576*x^8/8! + 1740438008*x^9/9! + 56893173354*x^10/10! + ...
%e A326009 such that
%e A326009 A(x) = 1 + (exp(2*x) + 1)*x + (exp(3*x) + 1)^2*x^2/2! + (exp(4*x) + 1)^3*x^3/3! + (exp(5*x) + 1)^4*x^4/4! + (exp(6*x) + 1)^5*x^5/5! + ...
%e A326009 also
%e A326009 A(x) = exp(x) + exp(2*x)*exp(exp(x)*x)*x + exp(6*x)*exp(exp(2*x)*x)*x^2/2! + exp(12*x)*exp(exp(3*x)*x)*x^3/3! + exp(20*x)*exp(exp(4*x)*x)*x^4/4! + ...
%o A326009 (PARI) /* E.g.f.: Sum_{n>=0} (1 + exp((n+1)*x))^n * x^n/n! */
%o A326009 {a(n) = my(A = sum(m=0, n, (1 + exp((m+1)*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A326009 for(n=0, 25, print1(a(n), ", "))
%o A326009 (PARI) /* E.g.f.: Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n/n! */
%o A326009 {a(n) = my(A = sum(m=0, n, exp(m*(m+1)*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
%o A326009 for(n=0, 25, print1(a(n), ", "))
%Y A326009 Cf. A326090, A326550.
%K A326009 nonn
%O A326009 0,2
%A A326009 _Paul D. Hanna_, Jul 13 2019