A362354 - OEIS

%I #16 May 28 2023 11:38:23

%S 1,3,15,108,1029,12288,177147,3000000,58461513,1289945088,31813498119,

%T 867763964928,25949267578125,844424930131968,29713734098717811,

%U 1124440102746243072,45543381089624394897,1966080000000000000000,90125827485245075684223,4372496892684322588065792

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

%C 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).

%C This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 3.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-function</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>

%F 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)).

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

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

%Y Cf. A137452.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Apr 24 2023