A326436 E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

1, 6, 95, 2307, 78000, 3433831, 188460821, 12508220886, 981371259995, 89426179550623, 9331384489007032, 1102143627943740931, 145924317814992561097, 21480095845779426077750, 3490477008130417972086807, 622292123277813938275834747, 121062971468108753273621477712, 25577093024015935514169919403295

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 = 4, r = 1.

Links

Table of n, a(n) for n=0..17.

Formula

E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

E.g.f.: exp(-5) * Sum_{n>=0} exp(n^2*x) * exp( 4*exp(n*x) ) / n!.

FORMULAS FOR TERMS.

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

Example

E.g.f.: A(x) = 1 + 6*x + 95*x^2/2! + 2307*x^3/3! + 78000*x^4/4! + 3433831*x^5/5! + 188460821*x^6/6! + 12508220886*x^7/7! + 981371259995*x^8/8! + 89426179550623*x^9/9! + 9331384489007032*x^10/10! + ...
such that
A(x) = exp(-5) * (1 + (exp(x) + 4) + (exp(2*x) + 4)^2/2! + (exp(3*x) + 4)^3/3! + (exp(4*x) + 4)^4/4! + (exp(5*x) + 4)^5/5! + (exp(6*x) + 4)^6/6! + ...)
also
A(x) = exp(-5) * (exp(4) + exp(x)*exp(4*exp(x)) + exp(4*x)*exp(4*exp(2*x))/2! + exp(9*x)*exp(4*exp(3*x))/3! + exp(16*x)*exp(4*exp(4*x))/4! + exp(25*x)*exp(4*exp(5*x))/5! + exp(36*x)*exp(4*exp(6*x))/6! + ...).

Prog

(PARI) /* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (exp(n*x +O(x^31)) + 4)^n/n! ) )))

Crossrefs

Cf. A326600, A020557, A326433, A326434, A326435, A326437.

Sequence in context: A058465 A290984 A338788 * A243802 A119627 A336825

Adjacent sequences: A326433 A326434 A326435 * A326437 A326438 A326439

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2019

Status

approved