A340838 a(n) = (1/2) * Sum_{k>=0} (k*(k + n))^n / 2^k.

1, 4, 139, 11928, 1909787, 491329088, 185373016419, 96425597012608, 66139668570414571, 57840395870803141632, 62813828698519808489915, 82933938539372018962724864, 130828514220436815006398809563, 243020960809424084526916839817216, 525038425527430196237626528753654867

Offset: 0

Author

Ilya Gutkovskiy, Jan 23 2021

Keywords

• nonn

Crossrefs

Cf. A000670, A098696, A249938, A292916, A340822.

Programs

Mathematica

Table[(1/2) Sum[(k (k + n))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Join[{1}, Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0] n^k, {k, 0, n}], {n, 1, 14}]]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000670(2*n-k) * n^k.