A140228 Binomial transform of [1, 5, 10, 10, 5, 1, 1, -1, 1, -1, 1, ...].

1, 6, 21, 56, 126, 252, 463, 798, 1308, 2058, 3129, 4620, 6650, 9360, 12915, 17506, 23352, 30702, 39837, 51072, 64758, 81284, 101079, 124614, 152404, 185010, 223041, 267156, 318066, 376536, 443387, 519498, 605808, 703318, 813093, 936264, 1074030, 1227660

Offset

0,2

Links

Zhuorui He, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

A007318 * [1, 5, 10, 10, 5, 1, 1, -1, 1, -1, 1, ...]

From Emeric Deutsch, Jun 03 2008: (Start)

a(n) = n*(274 + 85*n^2 + n^4)/60 for n >= 1. [corrected by Zhuorui He, Mar 01 2026]

G.f.: (1+x^6)/(1-x)^6. (End)

E.g.f.: 1 + exp(x)*x*(360 + 270*x + 110*x^2 + 10*x^3 + x^4)/60. - Stefano Spezia, Mar 02 2026

Example

a(4) = 56 = (1, 3, 3, 1) dot (1, 5, 10, 10) = (1 + 15 + 30 + 10).

Maple

1, seq((1/60)*n*(274+85*n^2+n^4), n=1..30); # Emeric Deutsch, Jun 03 2008

Mathematica

A140228[n_] := If[n == 0, 1, n*(274 + 85*n^2 + n^4)/60]; Array[A140228, 50, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 6, 21, 56, 126, 252, 463}, 50] (* Paolo Xausa, Apr 03 2026 *)

Crossrefs

Cf. A007318.

Sequence in context: A000389 A143980 A392547 * A264926 A006090 A192080

Adjacent sequences: A140225 A140226 A140227 * A140229 A140230 A140231

Keyword

nonn,easy

Author

Gary W. Adamson, May 12 2008

Extensions

More terms from Emeric Deutsch, Jun 03 2008

More terms from Zhuorui He, Mar 01 2026

Status

approved