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A326261 E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!.

%I A326261 #6 Jun 29 2019 10:55:45
%S A326261 1,4,18,115,1076,13749,223342,4437115,105308472,2930229721,
%T A326261 94110395546,3444510650343,142161931150564,6557368148307253,
%U A326261 335460464343013494,18907437932151629899,1167279375125285092592,78529603970775837111729,5730854443905658384812466,451803953552256670477653679,38337003901469883140928003036,3489532046271886600931347767373
%N A326261 E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!.
%C A326261 More generally, the following sums are equal:
%C A326261 (1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
%C A326261 (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A326261 here, q = exp(x) with p = 3, r = x.
%H A326261 Paul D. Hanna, <a href="/A326261/b326261.txt">Table of n, a(n) for n = 0..300</a>
%F A326261 E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n!.
%F A326261 E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x)*x ) * x^n/n!.
%e A326261 E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 115*x^3/3! + 1076*x^4/4! + 13749*x^5/5! + 223342*x^6/6! + 4437115*x^7/7! + 105308472*x^8/8! + 2930229721*x^9/9! + 94110395546*x^10/10! + ...
%e A326261 such that
%e A326261 A(x) = 1 + (3 + exp(x))*x + (3 + exp(2*x))^2*x^2/2! + (3 + exp(3*x))^3*x^3/3! + (3 + exp(4*x))^4*x^4/4! + (3 + exp(5*x))^5*x^5/5! + (3 + exp(6*x))^6*x^6/6! + ...
%e A326261 also
%e A326261 A(x) = exp(3*x) + exp(x + 3*exp(x)*x)*x + exp(4*x + 3*exp(2*x)*x)*x^2/2! + exp(9*x + 3*exp(3*x)*x)*x^3/3! + exp(16*x + 3*exp(4*x)*x)*x^4/4! + exp(25*x + 3*exp(5*x)*x)*x^5/5! + exp(36*x + 3*exp(6*x)*x)*x^6/6! + ...
%o A326261 (PARI) /* E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n! */
%o A326261 {a(n) = my(A = sum(m=0, n, (3 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A326261 for(n=0, 25, print1(a(n), ", "))
%o A326261 (PARI) /* E.g.f.: Sum_{n>=0} exp( n^2*x + 3*exp(n*x)*x ) * x^n/n! */
%o A326261 {a(n) = my(A = sum(m=0, n, exp(m^2*x + 3*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
%o A326261 for(n=0, 25, print1(a(n), ", "))
%Y A326261 Cf. A108459, A326090, A326091.
%K A326261 nonn
%O A326261 0,2
%A A326261 _Paul D. Hanna_, Jun 29 2019