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A326091 E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.

%I A326091 #9 Jun 29 2019 10:50:00
%S A326091 1,3,11,66,601,7418,116505,2248522,52025473,1414524690,44471074249,
%T A326091 1595792690594,64659403375137,2931455146804330,147550017664392457,
%U A326091 8189594420467104042,498288959815836863233,33061714451161940667554,2381086262720126177230473,185362512554618232339122578,15539467373234774634135507361,1398111233425766921500901239098,134584560980879138160145116701257
%N A326091 E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.
%C A326091 More generally, the following sums are equal:
%C A326091 (1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
%C A326091 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326091 here, q = exp(x) with p = 2, r = x.
%H A326091 Paul D. Hanna, <a href="/A326091/b326091.txt">Table of n, a(n) for n = 0..300</a>
%F A326091 E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n!.
%F A326091 E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x)*x ) * x^n/n!.
%e A326091 E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 66*x^3/3! + 601*x^4/4! + 7418*x^5/5! + 116505*x^6/6! + 2248522*x^7/7! + 52025473*x^8/8! + 1414524690*x^9/9! + 44471074249*x^10/10! + ...
%e A326091 such that
%e A326091 A(x) = 1 + (2 + exp(x))*x + (2 + exp(2*x))^2*x^2/2! + (2 + exp(3*x))^3*x^3/3! + (2 + exp(4*x))^4*x^4/4! + (2 + exp(5*x))^5*x^5/5! + (2 + exp(6*x))^6*x^6/6! + ...
%e A326091 also
%e A326091 A(x) = exp(2*x) + exp(x + 2*exp(x)*x)*x + exp(4*x + 2*exp(2*x)*x)*x^2/2! + exp(9*x + 2*exp(3*x)*x)*x^3/3! + exp(16*x + 2*exp(4*x)*x)*x^4/4! + exp(25*x + 2*exp(5*x)*x)*x^5/5! + exp(36*x + 2*exp(6*x)*x)*x^6/6! + ...
%o A326091 (PARI) /* E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n! */
%o A326091 {a(n) = my(A = sum(m=0, n, (2 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A326091 for(n=0, 25, print1(a(n), ", "))
%o A326091 (PARI) /* E.g.f.: Sum_{n>=0} exp( n^2*x + 2*exp(n*x)*x ) * x^n/n! */
%o A326091 {a(n) = my(A = sum(m=0, n, exp(m^2*x + 2*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
%o A326091 for(n=0, 25, print1(a(n), ", "))
%Y A326091 Cf. A108459, A326090, A326261.
%K A326091 nonn
%O A326091 0,2
%A A326091 _Paul D. Hanna_, Jun 28 2019