A061927 a(n) = n*(n+1)*(2*n+1)*(n^2+n+3)/30.

0, 1, 9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, 189202, 242802, 307923, 386331, 479964, 590940, 721565, 874341, 1051974, 1257382, 1493703, 1764303, 2072784, 2422992, 2819025

Offset: 0

Author

Henry Bottomley, May 17 2001

Keywords

• nonn
• easy

Comments

Also number of magic labelings of the cubical graph of magic sum n-1 [Ahmed]. - R. J. Mathar, Jan 25 2007

If Y_i (i=1,2,3) are 2-blocks of a (n+3)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007

The cube graph is also the prism graph I X C_4, so this is related to the number of magic labelings of other prism & related graphs. - David J. Seal, Sep 13 2017

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004, p. 73.
• Shalosh B. Ekhad and Doron Zeilberger, There are (1/30)(r+1)(r+2)(2r+3)(r^2+3r+5) Ways For the Four Teams of a World Cup Group to Each Have r Goals For and r Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1], arXiv:1407.1919 [math.CO], 2014.
• David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
• Yu-hong Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (5), m=3.
• Milan Janjic, Two Enumerative Functions
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 32.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006325, A019298, A244497, A244873, A289992, A292281, partial sums of A014820, A006975 (binomial transform shifted left).

Programs

Mathematica

Table[n (n + 1) (2 n + 1) (n^2 + n + 3)/30, {n, 0, 33}] (* or *)
CoefficientList[Series[x (1 + x)^3/(-1 + x)^6, {x, 0, 33}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,9,42,138,363},40] (* Harvey P. Dale, Apr 18 2018 *)

PARI

a(n) = { n*(n + 1)*(2*n + 1)*(n^2 + n + 3)/30 } \\ Harry J. Smith, Jul 29 2009

Formula

a(n) = a(n-1) + A014820(n) = A061926(9, n).

G.f.: x*(1+x)^3/(-1+x)^6 = 20/(-1+x)^5 + 1/(-1+x)^2 + 7/(-1+x)^3 + 18/(-1+x)^4 + 8/(-1+x)^6. - R. J. Mathar, Nov 18 2007