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

0,3

Comments

a(n)/n! is the invert transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

Links

Alois P. Heinz, Table of n, a(n) for n = 0..395

N. J. A. Sloane, Transforms

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (A001008(k)/A002805(k))*x^k).

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

Example

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! + ...

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}]

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ Seiichi Manyama, May 10 2023

Crossrefs

Cf. A000254, A001008, A002805, A007840, A128044, A128045, A305307.

Cf. A006153, A087761.

Sequence in context: A349515 A362356 A307679 * A113342 A262248 A346666

Adjacent sequences: A305303 A305304 A305305 * A305307 A305308 A305309

Keyword

nonn

Author

Ilya Gutkovskiy, May 29 2018

Status

approved