A362355 - OEIS

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A362355

a(n) = 4*(n+4)^(n-1).

1, 4, 24, 196, 2048, 26244, 400000, 7086244, 143327232, 3262922884, 82644187136, 2306601562500, 70368744177664, 2330488948919044, 83291859462684672, 3196026743131536484, 131072000000000000000, 5722274760967941313284, 264999811677837732610048

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This gives the fourth exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).

This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 4.

LINKS

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, Lambert W-function

Wikipedia, Lambert W function

FORMULA

a(n) = Sum_{k=0..n} |A137452(n, k)|*4^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*4^k, with the n = 0 term equal to 1 (not 0)).

E.g.f.: (LambertW(-x)/(-x))^4.

MATHEMATICA

Table[4(n+4)^(n-1), {n, 0, 20}] (* Harvey P. Dale, Jun 05 2024 *)

CROSSREFS

Column k = 4 of A232006.