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
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 05 2018
STATUS
approved