A232549 -id:A232549 - cp's OEIS frontend

A320082 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k/k!.

1, 1, 3, 5, -60, -186, 13832, -98862, -8631360, 352796880, 4245955032, -1185349047048, 48595690153920, 3201334718188320, -607575977909763840, 26489851912606455504, 4482546578798646251520, -958939334596403708474880, 50300999315063602037775360, 14223928928980522264922223360, -3933112779003946549567400925696

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A048994, A108459, A128943, A232549.

1,seq(n!*coeff(series(add(log(1+k*x)^k/k!, k=1..100), x=0, 21), x, n), n=1..20); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[1 + Sum[Log[1 + k x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^n, {k, n}], {n, 20}]]

a(n) = sum(k=0, n, stirling(n,k)*k^n); \\ Altug Alkan, Oct 05 2018

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^n.

A355375 a(n) = Sum_{k=0..n} (-k)^(n-k) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 0, -4, 10, 67, -969, 3341, 86976, -1988704, 14144108, 405611857, -17544321563, 287677263837, 3595470378748, -421298868094940, 14476946230894114, -112253861285434961, -18711849695261432065, 1354595712379990848137, -44436925726445545236496

Offset: 0

Seiichi Manyama, Jun 30 2022

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Cf. A229233, A232549, A318183, A355376.

a[n_] := Sum[(-k)^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (1-exp(-k*x))^k/(k^k*k!))))

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

cp's OEIS Frontend

A320082 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k/k!.

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A355375 a(n) = Sum_{k=0..n} (-k)^(n-k) * Stirling2(n,k).

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