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 A350366 #16 May 18 2025 04:10:12
%S A350366 1,2,18,312,8000,271770,11502162,583036832,34437042432,2322677883330,
%T A350366 176137593178250,14835018315726312,1373972097646792800,
%U A350366 138787120025382437882,15184417945878202716450,1788809909368939651651200,225755544056485027686459392
%N A350366 a(n) = [x^n] Product_{k=1..n} (1 + k*x)/(1 - k*x).
%F A350366 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n+1, k+1) * Stirling2(k+n, n).
%F A350366 a(n) ~ c * d^n * (n-1)!, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.281498742412700978029375818376931142913157133987685... - _Vaclav Kotesovec_, Dec 29 2021
%t A350366 a[n_] := Coefficient[Series[Product[(1 + k*x)/(1 - k*x), {k, 1, n}], {x, 0, n}], x, n]; Array[a, 17, 0] (* _Amiram Eldar_, Dec 27 2021 *)
%t A350366 Table[Sum[(-1)^(n - k)*StirlingS1[n + 1, k + 1] * StirlingS2[k + n, n], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 29 2021 *)
%o A350366 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*stirling(n+1, k+1, 1)*stirling(k+n, n, 2));
%Y A350366 Cf. A007820, A008275, A008277, A298851, A384043, A384044.
%K A350366 nonn
%O A350366 0,2
%A A350366 _Seiichi Manyama_, Dec 27 2021