%I #24 Jul 08 2019 23:46:22
%S 1,1,3,7,85,621,9391,153763,2985417,72857305,2023660891,64463660031,
%T 2355689179933,96585061458757,4411550668276935,223553495946198811,
%U 12468589636275471121,760686363827937889713,50544936457434070715827,3641703038385219667465975,283296615496743105794529381,23710312648334283124382200861,2128138462625913049294100820703
%N E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * exp(-x*(1+x)^n) * x^n / n!.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} (q^n + p)^n * exp(-p*q^n*r) * r^n/n!,
%C (2) Sum_{n>=0} (q^n - p)^n * exp(+p*q^n*r) * r^n/n!,
%C under suitable conditions; here, p = 1, q = 1+x, r = x.
%H Paul D. Hanna, <a href="/A326427/b326427.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * exp(-x*(1+x)^n) * x^n / n!.
%F E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * exp(+x*(1+x)^n) * x^n / n!.
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 7*x^3/3! + 85*x^4/4! + 621*x^5/5! + 9391*x^6/6! + 153763*x^7/7! + 2985417*x^8/8! + 72857305*x^9/9! + 2023660891*x^10/10! + ...
%e such that
%e A(x) = exp(-x) + ((1+x)+1)*exp(-x*(1+x))*x + ((1+x)^2+1)^2*exp(-x*(1+x)^2)*x^2/2! + ((1+x)^3+1)^3*exp(-x*(1+x)^3)*x^3/3! + ((1+x)^4+1)^4*exp(-x*(1+x)^4)*x^4/4! + ((1+x)^5+1)^5*exp(-x*(1+x)^5)*x^5/5! + ((1+x)^5+1)^6*exp(-x*(1+x)^6)*x^6/6! + ....
%e also
%e A(x) = exp(x) + ((1+x)-1)*exp(x*(1+x))*x + ((1+x)^2-1)^2*exp(x*(1+x)^2)*x^2/2! + ((1+x)^3-1)^3*exp(x*(1+x)^3)*x^3/3! + ((1+x)^4-1)^4*exp(x*(1+x)^4)*x^4/4! + ((1+x)^5-1)^5*exp(x*(1+x)^5)*x^5/5! + ((1+x)^5-1)^6*exp(x*(1+x)^6)*x^6/6! + ....
%e RELATED SERIES.
%e Below we illustrate the following identity at specific values of x:
%e Sum_{n>=0} ((1+x)^n + 1)^n * exp(-x*(1+x)^n) * x^n / n! = Sum_{n>=0} ((1+x)^n - 1)^n * exp(+x*(1+x)^n) * x^n / n!.
%e (1) At x = -1/2, the following sums are equal
%e S1 = Sum_{n>=0} (1 + 1/2^n)^n * exp(1/2^(n+1)) * (-1/2)^n / n!,
%e S1 = Sum_{n>=0} (1 - 1/2^n)^n * exp(-1/2^(n+1)) * (1/2^n) / n!,
%e where S1 = 0.87858219381273485321461030911688165612390011991090257894390...
%e (2) At x = -2/3, the following sums are equal
%e S2 = Sum_{n>=0} (1 + 1/3^n)^n * exp(2/3^(n+1)) * (-2/3)^n / n!,
%e S2 = Sum_{n>=0} (1 - 1/3^n)^n * exp(-2/3^(n+1)) * (2/3)^n / n!,
%e where S2 = 1.08434104867591642538828152304288635285016437420102640250465...
%o (PARI) {a(n) = my(A = sum(m=0,n, ((1+x)^m + 1 +x*O(x^n))^m * exp(-x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A = sum(m=0,n, ((1+x)^m - 1 +x*O(x^n))^m * exp(+x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A326428, A326425, A326426.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 04 2019