A305304 Expansion of e.g.f. 1/(1 + LambertW(-x/(1 + x))).

1, 1, 2, 9, 52, 405, 3786, 42301, 542984, 7924041, 129110230, 2327399481, 45940938924, 986045445853, 22856850513602, 569163515043285, 15150885843083536, 429364157810169105, 12905794670246364078, 410108007771441394129, 13736898888997174964660, 483740530150449507164901, 17866185834825657429606682

Offset

0,3

Comments

Inverse Lah transform of A000312.

Links

Table of n, a(n) for n=0..22.

N. J. A. Sloane, Transforms

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^k*n!/k!.

a(n) ~ n^n * (exp(1) - 1)^(n - 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Aug 18 2018

Maple

a:=series(1/(1+LambertW(-x/(1+x))), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 22; CoefficientList[Series[1/(1 + LambertW[-x/(1 + x)]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] k^k n!/k!, {k, n}], {n, 22}]]

Crossrefs

Cf. A000312, A052871, A060356, A305276.

Sequence in context: A248440 A330200 A143922 * A369090 A110322 A161631

Adjacent sequences: A305301 A305302 A305303 * A305305 A305306 A305307

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 18 2018

Status

approved