OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (1/2^n) * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(3*x/2) )^(1/3).
G.f.: 2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(2 - (3*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021
MATHEMATICA
a[n_] := (1/2^n) * Sum[(3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)
PROG
(PARI) a(n) = sum(k=0, n, (3*k+1)^(n-1)*binomial(n, k))/2^n;
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 26 2021
STATUS
approved