This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061927 #58 Dec 28 2024 00:06:34
%S A061927 0,1,9,42,138,363,819,1652,3060,5301,8701,13662,20670,30303,43239,
%T A061927 60264,82280,110313,145521,189202,242802,307923,386331,479964,590940,
%U A061927 721565,874341,1051974,1257382,1493703,1764303,2072784,2422992,2819025
%N A061927 a(n) = n*(n+1)*(2*n+1)*(n^2+n+3)/30.
%C A061927 Also number of magic labelings of the cubical graph of magic sum n-1 [Ahmed]. - _R. J. Mathar_, Jan 25 2007
%C A061927 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
%C A061927 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
%H A061927 Harry J. Smith, <a href="/A061927/b061927.txt">Table of n, a(n) for n = 0..1000</a>
%H A061927 Maya Mohsin Ahmed, <a href="https://arxiv.org/abs/math/0405476">Algebraic Combinatorics of Magic Squares</a>, arXiv:math/0405476 [math.CO], 2004, p. 73.
%H A061927 Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1407.1919">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]</a>, arXiv:1407.1919 [math.CO], 2014.
%H A061927 David Galvin and Courtney Sharpe, <a href="https://arxiv.org/abs/2409.15555">Independent set sequence of linear hyperpaths</a>, arXiv:2409.15555 [math.CO], 2024. See p. 7.
%H A061927 Yu-hong Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Guo/guo4.html">Some n-Color Compositions</a>, J. Int. Seq. 15 (2012) 12.1.2, eq. (5), m=3.
%H A061927 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A061927 Milan Janjic and Boris Petkovic, <a href="https://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A061927 Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H A061927 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973. [Cached copy, with permission] See p. 32.
%H A061927 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A061927 a(n) = a(n-1) + A014820(n) = A061926(9, n).
%F A061927 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
%t A061927 Table[n (n + 1) (2 n + 1) (n^2 + n + 3)/30, {n, 0, 33}] (* or *)
%t A061927 CoefficientList[Series[x (1 + x)^3/(-1 + x)^6, {x, 0, 33}], x] (* _Michael De Vlieger_, Sep 15 2017 *)
%t A061927 LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,9,42,138,363},40] (* _Harvey P. Dale_, Apr 18 2018 *)
%o A061927 (PARI) a(n) = { n*(n + 1)*(2*n + 1)*(n^2 + n + 3)/30 } \\ _Harry J. Smith_, Jul 29 2009
%Y A061927 Cf. A006325, A019298, A244497, A244873, A289992, A292281, partial sums of A014820, A006975 (binomial transform shifted left).
%K A061927 nonn,easy
%O A061927 0,3
%A A061927 _Henry Bottomley_, May 17 2001