A385752 -id:A385752 - cp's OEIS frontend

A385750 a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.

1, 1, 5, 64, 1681, 78651, 5891041, 653545390, 101785047169, 21431911982437, 5927319770834701, 2101574777340578156, 935265924020629176625, 512945332353359967175999, 341342159773993944429746793, 272012935493149854994361194426, 256689188247205271953044107166721, 284051735653584424779666013789038985

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

nonn

Cf. A000110, A064618, A119392, A119400, A385751, A385752.

Table[Sum[StirlingS2[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[(Exp[x] - 1)^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{k>=0} (x^k / k!^2) * Product_{j=1..k} 1 / (1 - j*x).

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

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

nonn

Cf. A119390, A320502, A385750, A385752.

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

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

cp's OEIS Frontend

A385750 a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula