A366151 a(n) = T(n, 3), where T(n, k) = Sum_{i=0..n} i^k * binomial(n, i) * (1/2)^(n-k).

0, 4, 20, 54, 112, 200, 324, 490, 704, 972, 1300, 1694, 2160, 2704, 3332, 4050, 4864, 5780, 6804, 7942, 9200, 10584, 12100, 13754, 15552, 17500, 19604, 21870, 24304, 26912, 29700, 32674, 35840, 39204, 42772, 46550

Offset: 0

Author

Peter Luschny, Oct 27 2023

Keywords

• nonn

Comments

A mean of binomials as might occur as the Expectation of random variables.

Crossrefs

T(n, 0) = A000012; T(n, 1) = A001477; T(n, 2) = A002378; T(n, 3) = this sequence.

T(1, n) = A011782; T(2, n) = A063376(n) (with offset 0); T(n, n) = A072034(n).

Programs

Maple

a := n -> n^2*(n + 3): seq(a(n), n = 0..35);

Formula

a(n) = n^2*(n + 3).

a(n) = [x^n] (2*x*(2 + 2*x - x^2))/(x - 1)^4.

a(n) = n! * [x^n] exp(x)*(x^3 + 6*x^2 + 4*x).