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

%I #16 Jun 21 2021 10:48:49

%S 1,1,5,28,353,3376,66637,908608,24405761,432891136,14712104501,

%T 321504185344,13218256749601,343360783937536,16565151205544957,

%U 498676704524517376,27614800115689879553,945381827279671853056,59095217374989483261925,2267322327322331161821184,157904201452248753415276001

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

%C 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.

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

%F a(n+1) = ((1 - n)^n + (1 + n)^n)/2. - _Stefano Spezia_, Jun 21 2021

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

%Y Cf. A345633 (odd part).

%Y Cf. A062024, A302583.

%Y Cf. A000169, A007778, A092364, A081131.

%K nonn,easy

%O 1,3

%A _Olivier GĂ©rard_, Jun 21 2021