A323619 Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.

1, 1, 0, 2, 3, 44, 260, 3534, 40796, 658440, 11066184, 220005840, 4750650432, 114430365048, 2993377996440, 85208541290040, 2611784941760640, 85941161628865344, 3018822193183216320, 112805065528683216192, 4467115744449046110720, 186900232401341222964480, 8237944325702047624948224

Offset: 0

Author

Ilya Gutkovskiy, Jan 20 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..398

Crossrefs

Cf. A000272, A038052, A048994, A277489, A305819.

Programs

Magma

[n le 0 select 1 else (&+[StirlingFirst(n,k)*k^(k-2): k in [1..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

Maple

seq(n!*coeff(series(1-LambertW(-log(1+x))*(2+LambertW(-log(1+x)))/2, x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019

Mathematica

nmax = 22; CoefficientList[Series[1 - LambertW[-Log[1 + x]] (2 + LambertW[-Log[1 + x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^(k - 2), {k, n}], {n, 22}]]

PARI

{a(n) = if(n==0,1, sum(k=1,n, stirling(n,k,1)*k^(k-2)))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

Sage

[1] + [sum((-1)^(k+n)*stirling_number1(n,k)*k^(k-2) for k in (1..n)) for n in (1..25)] # G. C. Greubel, Feb 07 2019

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000272(k).

a(n) ~ n^(n-2) / ((exp(exp(-1))-1)^(n - 3/2) * exp(n - 3*(1 - exp(-1))/2)). - Vaclav Kotesovec, Jan 20 2019