A326433 E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

1, 3, 29, 474, 11349, 366289, 15125300, 770762673, 47199596441, 3403242019876, 284281430425747, 27150503912943937, 2932403885598294838, 354869660881411722107, 47739034071736749352125, 7090201955561116768761250, 1155624866838027573814278801, 205611555585528308269669174557, 39746979329229607204823274477284

Offset

0,2

Comments

More generally, the following sums are equal:

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

FORMULAS FOR TERMS.

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

Example

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! + ...
such that
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! + ...)
also
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! + ...).

Prog

(PARI) /* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-2) * sum(n=0, 500, (exp(n*x +O(x^31)) + 1)^n/n! ) )))

Crossrefs

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

Sequence in context: A302923 A370922 A366005 * A113871 A186451 A248828

Adjacent sequences: A326430 A326431 A326432 * A326434 A326435 A326436

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2019

Status

approved