A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

1, 1, 5, 28, 353, 3376, 66637, 908608, 24405761, 432891136, 14712104501, 321504185344, 13218256749601, 343360783937536, 16565151205544957, 498676704524517376, 27614800115689879553, 945381827279671853056, 59095217374989483261925, 2267322327322331161821184, 157904201452248753415276001

Offset

1,3

Comments

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the even part.

Links

Table of n, a(n) for n=1..21.

Formula

a(n+1) = Sum_{k=0..floor(n/2)} n^(2k) binomial(n, 2k).

a(n+1) = ((1 - n)^n + (1 + n)^n)/2. - Stefano Spezia, Jun 21 2021

Mathematica

Table[Plus @@ Table[(n-1)^(2 k) Binomial[n-1, 2 k], {k, 0, Floor[n/2]}], {n, 1, 21}]

Crossrefs

Cf. A345633 (odd part).

Cf. A062024, A302583.

Cf. A000169, A007778, A092364, A081131.

Sequence in context: A174464 A372163 A354897 * A024068 A308593 A249784

Adjacent sequences: A345629 A345630 A345631 * A345633 A345634 A345635

Keyword

nonn,easy

Author

Olivier Gérard, Jun 21 2021

Status

approved