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

1, 2, 23, 480, 14627, 587580, 29331038, 1750923328, 121673580435, 9648709656300, 859874920598850, 85078769750118144, 9254316901029412110, 1097635452798476278232, 140986468651523106196060, 19496446561112852736019200, 2887977880849714395963280515

Offset: 0

Author

Seiichi Manyama, Dec 27 2021

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A007820, A008277, A129256, A298851, A350366, A383862, A384060.

Programs

Mathematica

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 *)
Table[Sum[StirlingS2[n + k, n]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 29 2021 *)

PARI

a(n) = sum(k=0, n, stirling(n+k, n, 2)*stirling(2*n-k, n, 2));

Formula

a(n) = Sum_{k=0..n} Stirling2(n+k, n) * Stirling2(2*n-k, n).

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