A362354 a(n) = 3*(n+3)^(n-1).

1, 3, 15, 108, 1029, 12288, 177147, 3000000, 58461513, 1289945088, 31813498119, 867763964928, 25949267578125, 844424930131968, 29713734098717811, 1124440102746243072, 45543381089624394897, 1966080000000000000000, 90125827485245075684223, 4372496892684322588065792

Offset: 0

Author

Wolfdieter Lang, Apr 24 2023

Keywords

• nonn
• easy

Comments

This gives the third 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 = 3.

Links

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

Crossrefs

Column k=3 of A232006 (without leading zeros).

Cf. A137452.

Formula

a(n) = Sum_{k=0..n} |A137452(n, k)|*3^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*3^k, with the n = 0 term equal to 1 (not 0).

E.g.f.: (LambertW(-x)/(-x))^3.

From Seiichi Manyama, Jun 19 2024: (Start)

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

(1) A(x) = exp(3*x*A(x)^(1/3)).

(2) A(x) = 1/A(-x*A(x)^(2/3)). (End)