A351333 -id:A351333 - cp's OEIS frontend

A351334 a(n) = Sum_{k=0..n} k! * (-k)^k * Stirling2(n,k).

1, -1, 7, -139, 5227, -317491, 28352347, -3495615859, 568791063547, -118065959980051, 30445266606199387, -9547490385298102579, 3578014749635903623867, -1579193384981544127824211, 810752966831581612807206427, -479049438742420410992820125299

Offset: 0

Seiichi Manyama, Feb 07 2022

sign

Cf. A122399, A351218, A351281, A351333.

a(n) = sum(k=0, n, k!*(-k)^k*stirling(n, k, 2));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(1-exp(x)))^k)))

E.g.f.: Sum_{k>=0} (k * (1 - exp(x)))^k.

A354674 a(n) = Sum_{k=0..n} k! * k^(k+n) * |Stirling1(n,k)|.

Original entry on oeis.org

1, 1, 33, 4568, 1653010, 1236180194, 1657339714418, 3620923498508952, 12037504737979759944, 57827877567223173191712, 385581993722741959459382352, 3454851578510897594456017095504, 40509304222426523176427339597382336

Offset: 0

Seiichi Manyama, Jun 02 2022

nonn

Cf. A320096, A350721, A350722, A351333, A351769.

a(n) = sum(k=0, n, k!*k^(k+n)*abs(stirling(n, k, 1)));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k*log(1-k*x))^k)))

E.g.f.: Sum_{k>=0} (-k * log(1 - k*x))^k.

cp's OEIS Frontend

A351334 a(n) = Sum_{k=0..n} k! * (-k)^k * Stirling2(n,k).

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A354674 a(n) = Sum_{k=0..n} k! * k^(k+n) * |Stirling1(n,k)|.

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Programs

PARI

PARI

Formula