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 A305051 #10 Feb 16 2025 08:33:54
%S A305051 1,2,12,119,1655,29647,649925,16852656,504519916,17124927207,
%T A305051 649856846635,27262957861405,1252893494644357,62593349657218070,
%U A305051 3377648236341185084,195782612085816693995,12131925601060324633027,800321307922970722566527,55998398887720317868148977
%N A305051 a(n) = n! * [x^n] exp(exp(x) - 1)/(1 - x)^n.
%H A305051 Vaclav Kotesovec, <a href="/A305051/b305051.txt">Table of n, a(n) for n = 0..360</a>
%H A305051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellNumber.html">Bell Number</a>
%H A305051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A305051 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A305051 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A305051 a(n) = ((-1)^n*n!/exp(1))*Sum_{k>=0} Laguerre(n,-2*n,k)/k!.
%F A305051 a(0) = 1; a(n) = (1/(n - 1)!)*Sum_{k=0..n} binomial(n,k)*(n + k - 1)!*Bell(n-k), where Bell() = A000110.
%F A305051 a(n) ~ c * n^n * 4^n / exp(n), where c = exp(exp(1/2) - 1)/sqrt(2) = 1.3527609882698012767793757868699146219161180684881726130481416807461987206887... - _Vaclav Kotesovec_, May 11 2021, updated Mar 18 2024
%t A305051 Table[n! SeriesCoefficient[Exp[Exp[x] - 1]/(1 - x)^n, {x, 0, n}], {n, 0, 18}]
%t A305051 Table[(-1)^n n!/Exp[1] Sum[LaguerreL[n,-2 n, k]/k!, {k, 0, Infinity}], {n, 0, 18}]
%t A305051 Join[{1}, Table[1/(n - 1)! Sum[Binomial[n, k] (n + k - 1)! BellB[n - k], {k, 0, n}], {n, 18}]]
%Y A305051 Cf. A000110, A101053, A101054, A101055.
%K A305051 nonn
%O A305051 0,2
%A A305051 _Ilya Gutkovskiy_, May 24 2018