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

Offset: 0

Author

Wolfdieter Lang, Apr 24 2023

Keywords

• nonn
• easy

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

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

Crossrefs

Column k = 4 of A232006 (without leading zeros).

Cf. A000272, A137452.

Programs

Mathematica

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

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.

From Seiichi Manyama, Jun 19 2024: (Start)

E.g.f. A(x) satisfies:

(1) A(x) = exp(4*x*A(x)^(1/4)).

(2) A(x) = 1/A(-x*A(x)^(1/2)). (End)