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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 29 2018
STATUS
approved