This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345633 #16 Jun 21 2021 10:49:35
%S A345633 0,1,4,36,272,4400,51012,1188544,18640960,567108864,11225320100,
%T A345633 421504185344,10079828372880,450353989316608,12627774819845668,
%U A345633 654244800082329600,21046391759976988928,1240529732459024678912,45032132922921758270916,2975557672677668838178816
%N A345633 Sum of terms of odd index in the binomial decomposition of n^(n-1).
%C A345633 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 odd part. See the Formula section.
%F A345633 a(n+1) = Sum_{k=0..floor((n-1)/2)} n^(2k+1)*binomial(n, 2k+1).
%F A345633 a(n+1) = ((1 + n)^n - (1 - n)^n)/2.
%t A345633 Table[Plus @@ Table[(n - 1)^(2 k + 1) Binomial[n - 1, 2 k + 1], {k, 0, Floor[(n - 1)/2]}], {n, 1, 21}]
%Y A345633 Cf. A345632 (even part).
%Y A345633 Cf. A062024, A302583.
%Y A345633 Cf. A000169, A007778, A092364, A081131.
%K A345633 nonn,easy
%O A345633 1,3
%A A345633 _Olivier Gérard_, Jun 21 2021