A384205 a(n) = [x^(2*n)] Product_{k=0..n} 1/(1 - k*x)^2.

1, 3, 201, 40792, 16904053, 11861321255, 12632193171300, 19003969060842360, 38387884967440214085, 100260769162534336491025, 328834941448280603509191681, 1323249839691864496146379353852, 6410573322270839015074278503521740, 36805304509116365389123823470306765972

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

a(n) = Sum_{k=0..2*n} Stirling2(n+k, n) * Stirling2(3*n-k, n).

a(n) ~ 2^(4*n - 1/2) * n^(2*n - 1/2) / (sqrt(Pi*(1-w)) * exp(2*n) * (2-w)^(2*n) * w^(2*n + 1/2)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

Mathematica

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

Crossrefs

Cf. A007820, A350376, A384206.

Sequence in context: A342578 A385834 A264674 * A306630 A140654 A229602

Adjacent sequences: A384202 A384203 A384204 * A384206 A384207 A384208

Keyword

nonn

Author

Vaclav Kotesovec, May 22 2025

Status

approved