A326009 E.g.f.: Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!.

1, 2, 8, 56, 564, 7452, 124126, 2527646, 61337576, 1740438008, 56893173354, 2116141180650, 88637462278492, 4144712080864292, 214742915441526686, 12247719772739219558, 764573919234220965072, 51977513845734053953776, 3830761480589037404767954, 304839727443701572462549058, 26096983659506717348854764356, 2395544800795092178844224643612, 235073598248121646307555752669446

Offset: 0

Author

Paul D. Hanna, Jul 13 2019

Keywords

• nonn

Comments

More generally, the following sums are equal:

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

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = exp(x), p = exp(-x), r = exp(x)*x.

Examples

			E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 564*x^4/4! + 7452*x^5/5! + 124126*x^6/6! + 2527646*x^7/7! + 61337576*x^8/8! + 1740438008*x^9/9! + 56893173354*x^10/10! + ...
such that
A(x) = 1 + (exp(2*x) + 1)*x + (exp(3*x) + 1)^2*x^2/2! + (exp(4*x) + 1)^3*x^3/3! + (exp(5*x) + 1)^4*x^4/4! + (exp(6*x) + 1)^5*x^5/5! + ...
also
A(x) = exp(x) + exp(2*x)*exp(exp(x)*x)*x + exp(6*x)*exp(exp(2*x)*x)*x^2/2! + exp(12*x)*exp(exp(3*x)*x)*x^3/3! + exp(20*x)*exp(exp(4*x)*x)*x^4/4! + ...
		

Crossrefs

Cf. A326090, A326550.

Programs

PARI

/* E.g.f.: Sum_{n>=0} (1 + exp((n+1)*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (1 + exp((m+1)*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

PARI

/* E.g.f.: Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m*(m+1)*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! equals the following sums.

(1) Sum_{n>=0} (exp((n+1)*x) + 1)^n * x^n / n!,

(2) Sum_{n>=0} exp(n*(n+1)*x) * exp(exp(n*x)*x) * x^n / n!.