A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).
1, 1, 5, 35, 324, 3744, 51902, 839362, 15513096, 322550616, 7451677632, 189366303840, 5249764639248, 157666361452560, 5099445234111888, 176713626295062384, 6531995374500741888, 256537368987293878272, 10667901271715707803264, 468261481657502075856768, 21635865693957558515860224
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 324*x^4/4! + 3744*x^5/5! + 51902*x^6/6! + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..395
- N. J. A. Sloane, Transforms
Programs
-
Maple
H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end: b:= proc(n) option remember; `if`(n=0, 1, add(H(j)*b(n-j), j=1..n)) end: a:= n-> b(n)*n!: seq(a(n), n=0..20); # Alois P. Heinz, May 29 2018 -
Mathematica
nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]! nmax = 20; CoefficientList[Series[1/(1 - Sum[HarmonicNumber[k] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]! a[0] = 1; a[n_] := a[n] = Sum[HarmonicNumber[k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}] -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ Seiichi Manyama, May 10 2023
Formula
a(n) ~ n! / ((1/LambertW(1)^2 - 1) * (1 - LambertW(1))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006153(k). - Seiichi Manyama, May 10 2023
Comments