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 A331690 #13 Jun 07 2022 10:57:08
%S A331690 1,1,4,33,456,9445,272448,10386817,503758720,30202999821,
%T A331690 2189000524800,188349613075393,18954958449853440,2203304642871358741,
%U A331690 292675996808408743936,44022321302156791898625,7438113993194856900034560,1401876939543892434209075581
%N A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).
%H A331690 Seiichi Manyama, <a href="/A331690/b331690.txt">Table of n, a(n) for n = 0..271</a>
%F A331690 a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
%F A331690 a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
%F A331690 a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
%F A331690 a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - _Vaclav Kotesovec_, Jun 06 2022
%t A331690 Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
%t A331690 Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
%t A331690 Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]
%o A331690 (PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ _Michel Marcus_, Jan 24 2020
%Y A331690 Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.
%K A331690 nonn
%O A331690 0,3
%A A331690 _Ilya Gutkovskiy_, Jan 24 2020