A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

1, 2, 4, 10, 30, 108, 444, 2112, 11040, 65712, 414816, 2992944, 21876816, 188936928, 1527813216, 15991733376, 133364903040, 1794144752640, 13329036288000, 270750383400960, 1167153128110080, 57074973648030720, -103080839984916480, 17319631144046423040, -171982551742151685120

Offset: 0

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

• sign

Comments

Exponential self-convolution of A006252.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005444, A005649, A006252, A052801, A089064.

Cf. A006252, A354120, A354121.

Programs

Maple

a:=series(1/(1 - log(1 + x))^2, x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 24; CoefficientList[Series[1/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (k + 1)!, {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*(k + 1)!.

a(n) ~ n! * 2 * (-1)^(n+1) / (n * log(n)^3) * (1 - 3*(gamma+1) / log(n) + (6*gamma^2 + 12*gamma + 6 - Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 15 2022

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023