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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..299
Programs
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Mathematica
Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *) -
PARI
a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));
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PARI
my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))
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