OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..372
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (1/2^n) * Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( x/LambertW( x * exp(-x) ) )^(1/2).
G.f.: 2 * Sum_{k>=0} (-2*k+1)^(k-1) * x^k/(2 - (-2*k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2 * exp(n) * LambertW(exp(-1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021
MATHEMATICA
a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 27 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, (-2*k+1)^(n-1)*binomial(n, k))/2^n;
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((x/lambertw(x*exp(-x)))^(1/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-2*k+1)^(k-1)*x^k/(2-(-2*k+1)*x)^(k+1)))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 27 2021
STATUS
approved