A289643 a(n) = n*(2*n+1)*binomial(n+2,n)/3.

0, 3, 20, 70, 180, 385, 728, 1260, 2040, 3135, 4620, 6578, 9100, 12285, 16240, 21080, 26928, 33915, 42180, 51870, 63140, 76153, 91080, 108100, 127400, 149175, 173628, 200970, 231420, 265205, 302560, 343728, 388960, 438515, 492660, 551670, 615828, 685425, 760760, 842140

Offset

0,2

Links

Table of n, a(n) for n=0..39.

Steve Butler and Pavel Karasik,, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, p=2 in the first displayed equation on page 4.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: -x*(3+5*x)/(x-1)^5.

a(n) = 3*A000332(n+3) + 5*A000332(n+2).

From Amiram Eldar, Jun 20 2025: (Start)

Sum_{n>=1} 1/a(n) = 23/2 - 16*log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*Pi - 4*log(2) - 19/2. (End)

Mathematica

Table[n(2n+1) Binomial[n+2, n]/3, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 20, 70, 180}, 40] (* Harvey P. Dale, May 18 2019 *)
Table[Sum[x^2 + y^2, {x, 0, g}, {y, x, g}], {g, 0, 39}] (* Horst H. Manninger, Jun 19 2025 *)

Crossrefs

Cf. A000332.

Sequence in context: A067600 A348208 A160456 * A196741 A196899 A006411

Adjacent sequences: A289640 A289641 A289642 * A289644 A289645 A289646

Keyword

nonn,easy

Author

R. J. Mathar, Jul 09 2017

Status

approved