A326433 - OEIS

%I #13 Jul 20 2019 16:33:27

%S 1,3,29,474,11349,366289,15125300,770762673,47199596441,3403242019876,

%T 284281430425747,27150503912943937,2932403885598294838,

%U 354869660881411722107,47739034071736749352125,7090201955561116768761250,1155624866838027573814278801,205611555585528308269669174557,39746979329229607204823274477284

%N E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

%C More generally, the following sums are equal:

%C (1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,

%C (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

%C here, q = exp(x), p = 1, r = 1.

%H Paul D. Hanna, <a href="/A326433/b326433.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

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

%F FORMULAS FOR TERMS.

%F a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

%e E.g.f.: A(x) = 1 + 3*x + 29*x^2/2! + 474*x^3/3! + 11349*x^4/4! + 366289*x^5/5! + 15125300*x^6/6! + 770762673*x^7/7! + 47199596441*x^8/8! + 3403242019876*x^9/9! + 284281430425747*x^10/10! + 27150503912943937*x^11/11! + 2932403885598294838*x^12/12! + ...

%e such that

%e A(x) = exp(-2) * (1 + (exp(x) + 1) + (exp(2*x) + 1)^2/2! + (exp(3*x) + 1)^3/3! + (exp(4*x) + 1)^4/4! + (exp(5*x) + 1)^5/5! + (exp(6*x) + 1)^6/6! + ...)

%e also

%e A(x) = exp(-2) * (exp(1) + exp(x)*exp(exp(x)) + exp(4*x)*exp(exp(2*x))/2! + exp(9*x)*exp(exp(3*x))/3! + exp(16*x)*exp(exp(4*x))/4! + exp(25*x)*exp(exp(5*x))/5! + exp(36*x)*exp(exp(6*x))/6! + ...).

%o (PARI) /* Requires suitable precision */

%o \p200

%o Vec(round(serlaplace( exp(-2) * sum(n=0,500, (exp(n*x +O(x^31)) + 1)^n/n! ) )))

%Y Cf. A326600, A020557, A326430, A326434, A326435, A326436, A326437.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 11 2019