A385751 -id:A385751 - 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.

A385752 a(n) = Sum_{k=0..n} Stirling1(n,k) * (n!/k!)^2.

Original entry on oeis.org

1, 1, -3, 46, -1967, 179351, -29861639, 8200834972, -3456505906559, 2118756407303197, -1811589861406160699, 2089746219541021377546, -3164800617505630505525903, 6151223064132377579849537011, -15052264342298428131766095419839, 45616620088948927404807879986431576, -168785206495071742797011703980958673919

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

sign

Cf. A119391, A192554, A385750, A385751.

Table[Sum[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.

cp's OEIS Frontend

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

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