cp's OEIS Frontend

A006325 4-dimensional analog of centered polygonal numbers.

%I A006325 #120 Sep 29 2024 04:32:43
%S A006325 0,0,1,7,26,70,155,301,532,876,1365,2035,2926,4082,5551,7385,9640,
%T A006325 12376,15657,19551,24130,29470,35651,42757,50876,60100,70525,82251,
%U A006325 95382,110026,126295,144305,164176,186032,210001,236215,264810,295926
%N A006325 4-dimensional analog of centered polygonal numbers.
%C A006325 If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 6-subsets of X intersecting both Y and Z. - _Milan Janjic_, Aug 26 2007
%C A006325 For n>0, a(n+1) is the n-th antidiagonal sum of A213751. - _Clark Kimberling_, Jun 20 2012
%C A006325 This sequence is the case m=n-1, k=n+3 of b(m,k) = m*(m+1)*((k-2)*m-(k-5))/6, which is the m-th k-gonal pyramidal number. - _Luciano Ancora_, Apr 11 2015
%C A006325 Starting (1, 7, 26, ...), this is the binomial transform of (1, 6, 13, 12, 4, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 31 2015
%C A006325 Also starting (1, 7, 26, ...), this appears to be the number of magic labelings of the cycle-of-loops graph LOOP X C_4 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. - _David J. Seal_, Sep 13 2017
%C A006325 The conjecture by David J. Seal is true and easily proved using MacMahon's Omega operators via the "Omega" package for Mathematica authored by Axel Riese (obtaining (up to an offset) the generating function listed in the formula section below). See the second Mathematica program in which the edges of LOOP X C_4 are indexed as in the example below. The Omega package can be downloaded from the link provided in the article by G. E. Andrews et al. - _L. Edson Jeffery_, Oct 15 2017
%D A006325 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H A006325 Vincenzo Librandi, <a href="/A006325/b006325.txt">Table of n, a(n) for n = 0..1000</a>
%H A006325 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis III. The Omega package</a>.
%H A006325 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 A006325 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A006325 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A006325 Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H A006325 Milan Janjic and B. 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 A006325 J. K. Merikoski, R. Kumar and R. A. Rajput, <a href="http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol26_pp168-176.pdf">Upper bounds for the largest eigenvalue of a bipartite graph</a>, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
%H A006325 Richard P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]
%H A006325 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A006325 a(n) = n*(n-1)*(n^2-n+1)/6.
%F A006325 a(n) = ((n^5 - (n-1)^5) - (n^1 - (n-1)^1))/30 = (n^5 - (n-1)^5 - 1)/30. - Xavier Acloque, Jan 25 2003
%F A006325 The partial sums of the octahedral numbers: a(n+1) = Sum_{i=0..n} A005900(i). - _Jonathan Vos Post_, Mar 14 2006
%F A006325 G.f.: -x^2*(x+1)^2/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F A006325 a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^2. - _Enrique Pérez Herrero_, Jan 15 2013 [Which is just rephrasing the partial sum formula with the Murthy formula in A005900. - _R. J. Mathar_, Jun 14 2014]
%F A006325 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - _Yosu Yurramendi_, Sep 03 2013
%F A006325 From _Daniel Poveda Parrilla_, Sep 09 2017: (Start)
%F A006325 a(n) = A169938(n-1)/6.
%F A006325 a(n+1) = A288486(n)/24. (End)
%F A006325 Sum_{n>=2} 1/a(n) = 12 - 2 * sqrt(3) * tanh(sqrt(3)*Pi/2). - _Amiram Eldar_, Jun 28 2020
%F A006325 E.g.f.: exp(x)*x^2*(3 + 4*x + x^2)/6. - _Stefano Spezia_, Dec 12 2021
%e A006325 A representation of the LOOP X C_4 graph, with edges and loops indexed as shown, as used in the second Mathematica program below:
%e A006325 .             3         1
%e A006325 .              O_______O
%e A006325 .              |   2   |
%e A006325 .              |4     0|
%e A006325 .              |_______|
%e A006325 .              O   6   O
%e A006325 .             5         7
%t A006325 Table[n*(n-1)*(n^2-n+1)/6, {n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 22 2011 *)
%t A006325 << Omega.m; n = 4; cond = {}; Do[AppendTo[cond, Sum[a[Mod[2*k - j, 2*n]], {j, 0, 2}] == a[2*n]], {k, 0, n - 1}]; f = OEqSum[Product[x[i]^a[i], {i, 0, 2*n}], cond, u][[1]]; Do[f = OEqR[f, Subscript[u, k]], {k, n}];
%t A006325 (* Generating function: *)
%t A006325 f = Factor[f /. {x[2*n] -> x} /. {x[_] -> 1}]
%t A006325 (* This sequence (with initial zeros dropped): *)
%t A006325 CoefficientList[Series[f, {x, 0, 35}], x] (* _L. Edson Jeffery_, Oct 15 2017 *)
%o A006325 (Magma) [n*(n-1)*(n^2-n+1)/6: n in [0..40]]; // _Vincenzo Librandi_, May 22 2011
%o A006325 (PARI) a(n)=n*(n-1)*(n^2-n+1)/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A006325 Cf. A005900, A169938, A213751, A288486, A019298, A061907, A244497, A244879, A244873, A244880.
%Y A006325 Cf. A000027, A000217, A019298, A244497, A244879, A244873, A244880, A293310, A293309 (magic labelings of LOOP X C_k, for k = 1..3,5..10).
%Y A006325 Cf. A293311, A293312.
%K A006325 nonn,easy
%O A006325 0,4
%A A006325 Albert Rich (Albert_Rich(AT)msn.com)