This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362524 #16 Aug 05 2025 07:01:38
%S A362524 1,1,2,4,16,56,391,2017,20504,139456,1867681,15751451,262263442,
%T A362524 2638794094,52589415971,614628436801,14274125637256,190012483804952,
%U A362524 5041005195499849,75288391385094811,2246914521052963166,37204717212894726706,1233884675800841217847
%N A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).
%H A362524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362524 E.g.f.: exp(x - LambertW(-x^2/2)) = -2 * LambertW(-x^2/2)/x^2 * exp(x).
%F A362524 a(n) ~ (exp(sqrt(2)*exp(-1/2) + 1) + (-1)^n*exp(1 - sqrt(2)*exp(-1/2))) * n^(n-1) / (2^((n-1)/2) * exp(n/2)). - _Vaclav Kotesovec_, Aug 05 2025
%t A362524 Table[n!Sum[(k+1)^(k-1)/(2^k k!(n-2k)!),{k,0,Floor[n/2]}],{n,0,25}] (* _Harvey P. Dale_, Mar 30 2024 *)
%o A362524 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^2/2))))
%Y A362524 Cf. A088957, A362525.
%Y A362524 Cf. A362350, A362522.
%K A362524 nonn,easy
%O A362524 0,3
%A A362524 _Seiichi Manyama_, Apr 23 2023