A128044 -id:A128044 - cp's OEIS frontend

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

Ilya Gutkovskiy, May 29 2018

nonn

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

			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! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..395
N. J. A. Sloane, Transforms

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

Cf. A006153, A087761.

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

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}]

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

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

A128045 a(n) = denominator of b(n), where b(1) = 1, b(n) = Sum_{k=1..n-1} b(n-k) * H(k); H(k) = Sum_{j=1..k} 1/j, the k-th harmonic number.

Original entry on oeis.org

1, 1, 2, 6, 2, 5, 360, 2520, 1680, 15120, 2700, 11880, 9979200, 8648640, 18345600, 2476656000, 27243216000, 714714000, 427508928000, 1160381376000, 1055947052160000, 22174888095360000, 38718058579200, 141031842336000

Offset: 1

Leroy Quet, Feb 11 2007

			1, 1, 5/2, 35/6, 27/2, 156/5, 25951/360, 419681/2520, 646379/1680, 13439609/15120, 5544403/2700, 56359019/11880, ...

Cf. A001008, A002805, A128044 (numerators), A305306.

f[l_List] := Block[{n = Length[l] + 1},Append[l, Sum[l[[n - k]]*HarmonicNumber[k], {k, n - 1}]]];Denominator[Nest[f, {1}, 24]] (* Ray Chandler, Feb 12 2007 *)

G.f. for fractions: x / (1 + log(1 - x) / (1 - x)). - Ilya Gutkovskiy, Sep 01 2021

Extended by Ray Chandler, Feb 12 2007

cp's OEIS Frontend

A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

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