OFFSET
0,5
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..411
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (1/2^n) * Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k).
E.g.f.: (x/2)/LambertW( x/2 * exp(-x/2) ).
G.f.: 2 * Sum_{k>=0} (-k+1)^(k-1) * x^k/(2 - (-k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Dec 05 2021
MATHEMATICA
a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 27 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;
(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))
(PARI) my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 27 2021
STATUS
approved