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, 50544, 54760, 59204, 63882, 68800

Offset

0,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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).

From Amiram Eldar, Nov 21 2025: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/18 - 11/54.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 - 2*log(2)/9 + 5/54. (End)

Maple

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

Mathematica

A366151[n_] := n^2*(n + 3); Array[A366151, 50, 0] (* Paolo Xausa, Nov 14 2025 *)

Prog

(PARI) a(n) = n^2*(n+3); \\ Amiram Eldar, Nov 21 2025

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).

Sequence in context: A250272 A023667 A213485 * A296274 A035007 A047810

Adjacent sequences: A366148 A366149 A366150 * A366152 A366153 A366154

Keyword

nonn,easy

Author

Peter Luschny, Oct 27 2023

Status

approved