A384206 a(n) = [x^(3*n)] Product_{k=0..n} 1/(1 - k*x)^2.

1, 4, 1291, 2107596, 9822847079, 99559982844000, 1870441451243408425, 58630795546429054116336, 2846132741588198942785663319, 202389763024999232451527049522000, 20194222519959431156536932169706390700, 2731878423936456763814384150978735866605108

Offset: 0

Author

Vaclav Kotesovec, May 22 2025

Keywords

• nonn

Crossrefs

Cf. A350376, A384207.

Programs

Mathematica

Table[SeriesCoefficient[Product[1/(1-k*x)^2, {k, 0, n}], {x, 0, 3*n}], {n, 0, 15}]
Table[Sum[StirlingS2[i+n, n] * StirlingS2[4*n-i, n], {i, 0, 3*n}], {n, 0, 15}]

Formula

a(n) = Sum_{k=0..3*n} Stirling2(n+k, n) * Stirling2(4*n-k, n).

a(n) ~ 5^(5*n + 1/2) * n^(3*n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 3/2) * exp(3*n) * w^(2*n + 1/2) * (5 - 2*w)^(3*n)), where w = -LambertW(-5*exp(-5/2)/2) = 0.268388115976977211740078521072609338...