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A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.

%I A326430 #11 Jul 11 2019 19:17:29
%S A326430 1,3,22,297,6055,169431,6145827,277912452,15225719420,988814989679,
%T A326430 74822364609113,6505084496930641,642317112612827029,
%U A326430 71331999557857791694,8835651007377368848464,1211946040741011512724559,182930472229597183037431011,30216143201862939999461382959,5435054718681965118312689633935
%N A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.
%C A326430 More generally, the following sums are equal:
%C A326430 (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
%C A326430 (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
%C A326430 here, q = exp(x), p = x, r = 1.
%H A326430 Paul D. Hanna, <a href="/A326430/b326430.txt">Table of n, a(n) for n = 0..300</a>
%F A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.
%F A326430 E.g.f.: exp(-1) * Sum_{n>=0} exp(n^2*x) * exp( x*exp(n*x) ) / n!.
%e A326430 E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 297*x^3/3! + 6055*x^4/4! + 169431*x^5/5! + 6145827*x^6/6! + 277912452*x^7/7! + 15225719420*x^8/8! + 988814989679*x^9/9! + 74822364609113*x^10/10! + ...
%e A326430 such that
%e A326430 A(x) = exp(-1) * (1 + (exp(x) + x) + (exp(2*x) + x)^2/2! + (exp(3*x) + x)^3/3! + (exp(4*x) + x)^4/4! + (exp(5*x) + x)^5/5! + (exp(6*x) + x)^6/6! + (exp(7*x) + x)^7/7! + (exp(8*x) + x)^8/8! + ...)
%e A326430 also,
%e A326430 A(x) = exp(-1) * (exp(x) + exp(x)*exp(x*exp(x)) + exp(4*x)*exp(x*exp(2*x))/2! + exp(9*x)*exp(x*exp(3*x))/3! + exp(16*x)*exp(x*exp(4*x))/4! + exp(25*x)*exp(x*exp(5*x))/5! + exp(36*x)*exp(x*exp(6*x))/6! + ...).
%o A326430 (PARI) /* Requires appropriate precision */
%o A326430 \p200
%o A326430 {a(n) = my(A = exp(-1) * sum(m=0,n+300, (exp(m*x +x*O(x^n)) + x)^m / m! )); round(n!*polcoeff(A,n))}
%o A326430 for(n=0,20,print1(a(n),", "))
%Y A326430 Cf. A326433.
%K A326430 nonn
%O A326430 0,2
%A A326430 _Paul D. Hanna_, Jul 09 2019