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A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.

%I A350376 #20 May 19 2025 04:43:13
%S A350376 1,2,23,480,14627,587580,29331038,1750923328,121673580435,
%T A350376 9648709656300,859874920598850,85078769750118144,9254316901029412110,
%U A350376 1097635452798476278232,140986468651523106196060,19496446561112852736019200,2887977880849714395963280515
%N A350376 a(n) = [x^n] Product_{k=1..n} 1/(1 - k*x)^2.
%H A350376 Vaclav Kotesovec, <a href="/A350376/b350376.txt">Table of n, a(n) for n = 0..300</a>
%F A350376 a(n) = Sum_{k=0..n} Stirling2(n+k, n) * Stirling2(2*n-k, n).
%F A350376 a(n) ~ c * d^n * (n-1)!, where d = 27 / (4*LambertW(-3*exp(-3/2)/2)^2 * (3 + 2*LambertW(-3*exp(-3/2)/2))) = 9.858422414446789720857925020919293523149... and c = sqrt(3/(-LambertW(-3*exp(-3/2)/2) * (1 + LambertW(-3*exp(-3/2)/2)))) / (4*Pi) = 0.28482428628793763109169664913715827647091747... - _Vaclav Kotesovec_, Dec 28 2021, updated May 14 2025
%t A350376 a[n_] := Coefficient[Series[Product[1/(1 - k*x)^2, {k, 1, n}], {x, 0, n}], x, n]; Array[a, 17, 0] (* _Amiram Eldar_, Dec 28 2021 *)
%t A350376 Table[Sum[StirlingS2[n + k, n]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Dec 29 2021 *)
%o A350376 (PARI) a(n) = sum(k=0, n, stirling(n+k, n, 2)*stirling(2*n-k, n, 2));
%Y A350376 Cf. A007820, A008277, A129256, A298851, A350366, A383862, A384060.
%K A350376 nonn
%O A350376 0,2
%A A350376 _Seiichi Manyama_, Dec 27 2021