A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..299

Formula

a(n) = Sum_{k=0..n} k^(2*n-k).

a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

Mathematica

Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)

Prog

(PARI) a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))

Crossrefs

Cf. A031971, A349883.

Cf. A249459, A349884, A368505.

Sequence in context: A195242 A243697 A106273 * A052803 A201923 A222059

Adjacent sequences: A349833 A349834 A349835 * A349837 A349838 A349839

Keyword

nonn

Author

Seiichi Manyama, Dec 03 2021

Status

approved