A362356 a(n) = 5*(n + 5)^(n-1).

1, 5, 35, 320, 3645, 50000, 805255, 14929920, 313742585, 7378945280, 192216796875, 5497558138880, 171359481538165, 5784156907130880, 210264917311285295, 8192000000000000000, 340611592914758411505, 15056807481695325716480, 705250197803314844630515

Offset: 0

Author

Wolfdieter Lang, Apr 24 2023

Keywords

• nonn
• easy

Comments

This gives the fifth 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 = 5.

Links

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

Crossrefs

Column k=5 of A232006 (without leading zeros).

Cf. A000272, A137452.

Formula

a(n) = Sum_{k=0..n} |A137452(n, k)|*5^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*5^k, with the n = 0 term equal to 1 (not 0).

E.g.f.: (LambertW(-x)/(-x))^5.

From Seiichi Manyama, Jun 19 2024: (Start)

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

(1) A(x) = exp(5*x*A(x)^(1/5)).

(2) A(x) = 1/A(-x*A(x)^(2/5)). (End)