A363397 a(n) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} binomial(k, j) * (j + 1)^n. Row sums of A363399.

1, 5, 32, 302, 3904, 64272, 1286144, 30313712, 822571008, 25258008320, 865863532544, 32779942009344, 1358320701014016, 61149815860711424, 2971951570679234560, 155090406558662064128, 8649258967534890123264, 513370937392454603833344

Offset: 0

Author

Peter Luschny, Jun 02 2023

Keywords

• nonn

Crossrefs

Cf. A363399.

Programs

Maple

a := n -> add(add(binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):
seq(a(n), n = 0..17);

Mathematica

Table[Sum[2^(n-k) * Sum[Binomial[k, j] * (j+1)^n, {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 02 2023 *)

Formula

a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^n / ((1 - LambertW(exp(-1))) * exp(n) * LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Jun 02 2023