OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..398
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = -log(1-x)/LambertW(-log(1-x)).
a(n) ~ -(-1)^n * n^(n-1) / ((exp(exp(-1)) - 1)^(n - 1/2) * exp(n + exp(-1)/2 + 1/2)). - Vaclav Kotesovec, Nov 22 2021
EXAMPLE
A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ... = 1 + x + x^2 + x^3 + ... = 1/(1 - x).
MATHEMATICA
Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 22 2021 *)
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 1));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (k-1)^(k-1)*log(1-x)^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-x)/lambertw(-log(1-x))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 22 2021
STATUS
approved