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A302581
a(n) = n! * [x^n] -exp(-n*x)*log(1 - x).
1
0, 1, -3, 20, -186, 2249, -33360, 586172, -11901008, 274098393, -7060189120, 201092672604, -6275340884736, 212915635727313, -7803567334571008, 307245946117223700, -12933084380738398208, 579587518114690731601, -27550568677612746940416, 1384553892443352890245636
OFFSET
0,3
FORMULA
a(n) = Sum_{k=1..n} (-n)^(n-k)*(k-1)!*binomial(n,k).
E.g.f.: -log(1 - LambertW(x))/(1 + LambertW(x)). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ -(-1)^n * log(2) * n^n. - Vaclav Kotesovec, Jun 09 2019
MATHEMATICA
Table[n! SeriesCoefficient[-Exp[-n x] Log[1 - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-n)^(n - k) (k - 1)! Binomial[n, k], {k, 1, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[-Log[1 - LambertW[x]]/(1 + LambertW[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Apr 10 2018
STATUS
approved