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%I #15 Jul 13 2019 11:56:55
%S 1,2,6,35,308,3637,55150,1033027,23260536,617066297,18968614874,
%T 666664879663,26496140541700,1179815542970053,58388906382906390,
%U 3189604848766578563,191168734534622234480,12504288586619417431921,888401197086798248554546,68270033412187747029025111,5652853046029263008213465916,502601914954325406783531231677,47834047958592244085651443711406
%N E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
%C (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C here, q = exp(x) with p = 1, r = x.
%H Paul D. Hanna, <a href="/A326090/b326090.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!.
%F E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.
%e E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 35*x^3/3! + 308*x^4/4! + 3637*x^5/5! + 55150*x^6/6! + 1033027*x^7/7! + 23260536*x^8/8! + 617066297*x^9/9! + 18968614874*x^10/10! + ...
%e such that
%e A(x) = 1 + (1 + exp(x))*x + (1 + exp(2*x))^2*x^2/2! + (1 + exp(3*x))^3*x^3/3! + (1 + exp(4*x))^4*x^4/4! + (1 + exp(5*x))^5*x^5/5! + (1 + exp(6*x))^6*x^6/6! + ...
%e also
%e A(x) = exp(x) + exp(x + exp(x)*x)*x + exp(4*x + exp(2*x)*x)*x^2/2! + exp(9*x + exp(3*x)*x)*x^3/3! + exp(16*x + exp(4*x)*x)*x^4/4! + exp(25*x + exp(5*x)*x)*x^5/5! + exp(36*x + exp(6*x)*x)*x^6/6! + ...
%e RELATED SERIES.
%e Below we illustrate the following identity at specific values of x:
%e Sum_{n>=0} (1 + exp(n*x))^n * x^n/n! = Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.
%e (1) At x = -1, the following sums are equal
%e S1 = Sum_{n>=0} (1 + exp(-n))^n * (-1)^n/n!,
%e S1 = Sum_{n>=0} exp(-n^2) * exp( -exp(-n) ) * (-1)^n/n!,
%e where S1 = 0.12121214669421724219987424741512642137552627624687959194...
%e (2) At x = -log(2), the following sums are equal
%e S2 = Sum_{n>=0} (1 + 1/2^n)^n * log(1/2)^n/n!,
%e S2 = Sum_{n>=0} 2^(-n^2) * 2^(-1/2^n) * log(1/2)^n/n!,
%e where S2 = 0.26746154600304489791062659014323146833150028333177021587...
%o (PARI) /* E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n! */
%o {a(n) = my(A = sum(m=0, n, (1 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* E.g.f.: Sum_{n>=0} exp( n^2*x + exp(n*x)*x ) * x^n/n! */
%o {a(n) = my(A = sum(m=0, n, exp(m^2*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A108459, A326091, A326261, A326009.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 28 2019
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