A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

1, 1, 9, 212, 9418, 675014, 71092502, 10334690232, 1982433606264, 485065343565072, 147433546709109408, 54493722609862927632, 24069397682825072219040, 12520250948941157091235344, 7575515622713954399390221008, 5275250174853125498317783254528

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..229

Formula

a(n) ~ c * d^n * n^(2*n + 1/2), where

w = -LambertW(-1, -exp(-r)*r) = 1.1628296650659469964248518258036278907318113...

r = 0.8531304407911771560472963194514988627832723535823134189532... is the real root of the equation w = r + exp(-1/r)

d = exp(-1)*r*w*(w-r)^(r-1) = 0.433513333588184444899487502412976956849408575992...

c = 1.959633090979666812031505093625147349925787002426082...

E.g.f.: Sum_{k>=0} (-log(1 - k*x))^k. - Seiichi Manyama, Feb 02 2022

Mathematica

Flatten[{1, Table[Sum[(-1)^(n-k)*StirlingS1[n, k]*k!*k^n, {k, 1, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 04 2022 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^n*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 02 2022
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k*x))^k))) \\ Seiichi Manyama, Feb 02 2022

Crossrefs

Cf. A007840, A220181, A320083.

Sequence in context: A300136 A274672 A122399 * A327198 A188409 A109587

Adjacent sequences: A320093 A320094 A320095 * A320097 A320098 A320099

Keyword

nonn

Author

Vaclav Kotesovec, Oct 05 2018

Status

approved