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A362355 a(n) = 4*(n+4)^(n-1).

%I A362355 #26 Jun 19 2025 05:57:54
%S A362355 1,4,24,196,2048,26244,400000,7086244,143327232,3262922884,
%T A362355 82644187136,2306601562500,70368744177664,2330488948919044,
%U A362355 83291859462684672,3196026743131536484,131072000000000000000,5722274760967941313284,264999811677837732610048
%N A362355 a(n) = 4*(n+4)^(n-1).
%C A362355 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).
%C A362355 This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 4.
%H A362355 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-function</a>
%H A362355 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F A362355 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).
%F A362355 E.g.f.: (LambertW(-x)/(-x))^4.
%F A362355 From _Seiichi Manyama_, Jun 19 2024: (Start)
%F A362355 E.g.f. A(x) satisfies:
%F A362355 (1) A(x) = exp(4*x*A(x)^(1/4)).
%F A362355 (2) A(x) = 1/A(-x*A(x)^(1/2)). (End)
%t A362355 Table[4(n+4)^(n-1),{n,0,20}] (* _Harvey P. Dale_, Jun 05 2024 *)
%Y A362355 Column k = 4 of A232006 (without leading zeros).
%Y A362355 Cf. A000272, A137452.
%K A362355 nonn,easy
%O A362355 0,2
%A A362355 _Wolfdieter Lang_, Apr 24 2023