%I A002817 M4141 N1718
%S A002817 0,1,6,21,55,120,231,406,666,1035,1540,2211,3081,4186,5565,7260,9316,
%T A002817 11781,14706,18145,22155,26796,32131,38226,45150,52975,61776,71631,
%U A002817 82621,94830,108345,123256,139656,157641,177310,198765,222111,247456
%N A002817 Doubly triangular numbers: n*(n+1)*(n^2+n+2)/8.
%C A002817 Number of inequivalent ways to color vertices of a square using <= n 
               colors, allowing rotations and reflections. Group is dihedral group 
               D_8 of order 8 with cycle index (1/8)*(x1^4+2*x4+3*x2^2+2*x1^2*x2); 
               setting all x_i = n gives the formula a(n) = (1/8)*(n^4+2*n+3*n^2+2*n^3).
%C A002817 Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 
               X 3 matrices of nonnegative integers such that row sums and column 
               sums are all equal to n-1. - Peter Bertok (peter(AT)bertok.com), 
               Jan 12 2002. See A005045 for another version.
%C A002817 Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 
               + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex 
               of the cycle matroid of the complete graph K_n on n vertices (n>=2) 
               - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006
%C A002817 If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to 
               the number of 5-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), 
               Jul 30 2007
%C A002817 Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 
               0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 05 2009]
%D A002817 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002817 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002817 A. Bj\"orner, The homology and shellability of matroids and geometric 
               lattices, in Matroid Applications (ed. N. White), Encyclopedia of 
               Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.
%D A002817 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
%D A002817 I. J. Good, On the application of symmetric Dirichlet distributions and 
               their mixtures to contingency tables. Ann. Statist. 4 (1976), no. 
               6, 1159-1189.
%D A002817 D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double 
               stochastic nonnegative integer square matrices, SIAM J. Comput., 
               4 (1975), 474-477.
%D A002817 R. P. Stanley, Enumerative Combinatorics I, p. 292.
%D A002817 Warburton, Henry. "On Self-Repeating Series." Transactions of the Cambridge 
               Philosophical Society, Vol. 9, 471-486, 1856.
%H A002817 T. D. Noe, <a href="b002817.txt">Table of n, a(n) for n=0..1000</a>
%H A002817 G. E. Andrews, P. Paule, A. Riese and V. Strehl, <a href="http://www.risc.uni-linz.ac.at/
               research/combinat/risc/publications/#ppaule">MacMahon's partition 
               analysis V. Bijections, recursions and magic squares</a>, p. 37.
%H A002817 Matthias Beck, <a href="http://arXiv.org/abs/math.CO/0201013">The number 
               of "magic" squares and hypercubes</a>
%H A002817 P. Diaconis and A. Gamburd, <a href="http://www.combinatorics.org/Volume_11/
               Abstracts/v11i2r2.html">Random matrices, magic squares and matching 
               polynomials</a>
%H A002817 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A002817 Neven Juric, <a href="a002817.ps">Illustration of the 55 3 X 3 matrices</
               a>
%H A002817 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002817 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A002817 G.f.: x(1+x+x^2)/(1-x)^5. a(n) = 3*binomial(n+2, 4)+binomial(n+1, 2).
%F A002817 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3 - Warut Roonguthai 
               (warut822(AT)yahoo.com) Dec 13 1999
%F A002817 a(n) = Sum [ Sum ( 1 + Sum (3*n) ) ]. - Xavier Acloque, Jan 21 2003
%F A002817 a(n) = (n+3 choose 4) + (n+2 choose 4) + (n+1 choose 4) - Mitch Harris 
               (Harris.Mitchell(AT)mgh.harvard.edu), Oct 17 2006
%p A002817 A002817 := n->n*(n+1)*(n^2+n+2)/8;
%p A002817 A002817:=-(1+z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation.]
%p A002817 a:=n->add(n+add(binomial(n,2), j=1..n),j=2..n):seq(a(n)/4, n=1..38); 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
%t A002817 a[n_] := n(n+1)(n^2+n+2)/8
%o A002817 (PARI) a(n)=n*(n+1)*(n^2+n+2)/8
%Y A002817 Cf. A000217, A064322, A066370, A001496.
%Y A002817 A001621 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
%Y A002817 Sequence in context: A067680 A115052 A025203 this_sequence A132366 A015641 
               A050190
%Y A002817 Adjacent sequences: A002814 A002815 A002816 this_sequence A002818 A002819 
               A002820
%K A002817 nonn,easy,nice
%O A002817 0,3
%A A002817 N. J. A. Sloane (njas(AT)research.att.com).
%E A002817 More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), 
               Dec 29 1999
%E A002817 Plouffe Maple line edited by N. J. A. Sloane (njas(AT)research.att.com), 
               May 13 2008