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, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..410
N. J. A. Sloane, Transforms
FORMULA
a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...
MAPLE
g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
add(g(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 = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 29 2018
STATUS
approved