A385751 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (n!/k!)^2.

1, 1, 5, 100, 5137, 539851, 101035441, 30669875230, 14117057058945, 9364637252286181, 8603755430968248301, 10603853731438585516856, 17077610933602804111318705, 35160631271792580418277658415, 90839446923946068488317221868825, 289828370988497912073923950177143826, 1126236403418687405801564385561640043521

Offset: 0

Author

Ilya Gutkovskiy, Jul 08 2025

Keywords

• nonn

Crossrefs

Cf. A119390, A320502, A385750, A385752.

Programs

Mathematica

Table[Sum[Abs[StirlingS1[n, k]] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Sum[(-Log[1 - x])^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (-log(1 - x))^k / k!^3.

a(n) ~ (sqrt(2*Pi/3) * exp(3*log(n)^(1/3) - 3*n) * n^(3*n + 1/2) / log(n)) * (1 - 2/(9*log(n)^(1/3)) + (gamma - 4/81)/log(n)^(2/3) - (40/2187 + 11*gamma/9)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 09 2025