A350366 - OEIS

%I #14 Dec 29 2021 04:03:20

%S 1,2,18,312,8000,271770,11502162,583036832,34437042432,2322677883330,

%T 176137593178250,14835018315726312,1373972097646792800,

%U 138787120025382437882,15184417945878202716450,1788809909368939651651200,225755544056485027686459392

%N a(n) = [x^n] Product_{k=1..n} (1 + k*x)/(1 - k*x).

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n+1, k+1) * Stirling2(k+n, n).

%F 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 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 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 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*stirling(n+1, k+1, 1)*stirling(k+n, n, 2));

%Y Cf. A007820, A008275, A008277, A298851.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 27 2021