A350366 a(n) = [x^n] Product_{k=1..n} (1 + k*x)/(1 - k*x).
1, 2, 18, 312, 8000, 271770, 11502162, 583036832, 34437042432, 2322677883330, 176137593178250, 14835018315726312, 1373972097646792800, 138787120025382437882, 15184417945878202716450, 1788809909368939651651200, 225755544056485027686459392
Offset: 0
Keywords
Programs
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Mathematica
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 *) 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 *) -
PARI
a(n) = sum(k=0, n, (-1)^(n-k)*stirling(n+1, k+1, 1)*stirling(k+n, n, 2));
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n+1, k+1) * Stirling2(k+n, n).
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