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A320096
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.
10
1, 1, 9, 212, 9418, 675014, 71092502, 10334690232, 1982433606264, 485065343565072, 147433546709109408, 54493722609862927632, 24069397682825072219040, 12520250948941157091235344, 7575515622713954399390221008, 5275250174853125498317783254528
OFFSET
0,3
LINKS
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