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 A005130 Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's). (Formerly M1808) 45

%I M1808

%S 1,1,2,7,42,429,7436,218348,10850216,911835460,129534272700,

%T 31095744852375,12611311859677500,8639383518297652500,

%U 9995541355448167482000,19529076234661277104897200,64427185703425689356896743840,358869201916137601447486156417296

%N Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's).

%C Also known as the Andrews-Mills-Robbins-Rumsey numbers. - _N. J. A. Sloane_, May 24 2013

%C An alternating sign matrix is a matrix of 0's, 1's and -1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.

%C From _Gary W. Adamson_, May 27 2009: (Start)

%C Starting with offset 1 = row sums of triangle A160708, and convolution square of A160707.

%C a(n) is odd iff n is a Jacobsthal number [Frey and Sellers, 2000].

%C Starting with offset 1 = row sums of triangle A160708.

%C Starting (1, 2, 7,...) = convolution square of A160707: [1, 1, 3, 18, 192,...].

%C (End)

%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.

%D C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386.

%D C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005130/b005130.txt">Table of n, a(n) for n = 0..100</a>

%H T. Amdeberhan, V. H. Moll, <a href="http://www.emis.ams.org/journals/EJC/Volume_18/Abstracts/v18i2p1.html">Arithmetic properties of plane partitions</a>, El. J. Comb. 18 (2) (2011) # P1.

%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/BF01389763">Plane partitions (III): the Weak Macdonald Conjecture</a>, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.

%H M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="https://arxiv.org/abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions</a>, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001.

%H E. Beyerstedt, V. H. Moll, X. Sun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Moll/moll2.html">The p-adic Valuation of the ASM Numbers</a>, J. Int. Seq. 14 (2011) # 11.8.7

%H D. M. Bressoud and J. Propp, <a href="http://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.

%H H. Cheballah, S. Giraudo, R. Maurice, <a href="https://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.

%H M. Ciucu, <a href="http://dx.doi.org/10.1006/jcta.1998.2922">The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions</a>, J. Combin. Theory Ser. A 86 (1999), 382-389.

%H F. Colomo and A. G. Pronko, <a href="http://arXiv.org/abs/math-ph/0404045">On the refined 3-enumeration of alternating sign matrices</a>, Advances in Applied Mathematics 34 (2005) 798.

%H F. Colomo and A. G. Pronko, <a href="https://arxiv.org/abs/math-ph/0411076">Square ice, alternating sign matrices and classical orthogonal polynomials</a>, JSTAT (2005) P01005.

%H G. Conant, <a href="http://homepages.math.uic.edu/~gconant/Math/magmas.pdf">Magmas and Magog Triangles</a>, 2014.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/AlternatingSignMatrices">Alternating sign matrices</a>

%H I. Fischer, <a href="https://arxiv.org/abs/math/0501102">The number of monotone triangles with prescribed bottom row</a>, arXiv:math/0501102 [math.CO], 2005.

%H T. Fonseca, F. Balogh, <a href="http://dx.doi.org/10.1007/s10801-014-0555-0">The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials</a>, Journal of Algebraic Combinatorics, 2014, <a href="http://arxiv.org/abs/1210.4527">arXiv:1210.4527</a>

%H P. Di Francesco, <a href="https://arxiv.org/abs/cond-mat/0409576">A refined Razumov-Stroganov conjecture II</a>, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.

%H P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/math-ph/0410002">Determinant formulas for some tiling problems...</a>, arXiv:math-ph/0410002, 2004.

%H D. D. Frey and J. A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/sellers.pdf">Jacobsthal Numbers and Alternating Sign Matrices</a>, Journal of Integer Sequences Vol. 3 (2000) #00.2.3.

%H D. D. Frey and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/p23.pdf">Prime Power Divisors of the Number of n X n Alternating Sign Matrices</a>

%H M. Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991.

%H J. de Gier, <a href="https://arxiv.org/abs/math/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002-2003.

%H C. Heuberger, H. Prodinger, <a href="http://dx.doi.org/10.1142/S1793042111003892">A precise description of the p-adic valuation of the number of alternating sign matrices</a>, Intl. J. Numb. Th. 7 (1) (2011) 57-69

%H Heuer, Dylan, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also <a href="http://arxiv.org/abs/1611.03387">arXiv:1611.03387</a>.

%H G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/9712207">Another proof of the alternating-sign matrix conjecture</a>, Internat. Math. Res. Notices, No. 3, (1996), 139-150.

%H G. Kuperberg, <a href="https://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv math.CO/0008184, <a href="http://dx.doi.org/10.2307/3597283">Ann. Math. 156 (3) (2002) 835-866</a>

%H W. H. Mills, David P Robbins, Howard Rumsey Jr., <a href="http://dx.doi.org/10.1016/0097-3165(83)90068-7">Alternating sign matrices and descending plane partitions</a> J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013)

%H Igor Pak, <a href="https://arxiv.org/abs/1803.06636">Complexity problems in enumerative combinatorics</a>, arXiv:1803.06636 [math.CO], 2018.

%H C. Pickover, <a href="/A005130/a005130_2.pdf">Mazes for the Mind</a>, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386. [Annotated scanned copy]

%H J. Propp, <a href="http://www.dmtcs.org/pdfpapers/dmAA0103.pdf">The many faces of alternating-sign matrices.</a>

%H A. V. Razumov and Yu. G. Stroganov, <a href="http://arXiv.org/abs/cond-mat/0012141">Spin chains and combinatorics</a>, arXiv cond-mat/0012141

%H Lukas Riegler, <a href="http://homepage.univie.ac.at/lukas.riegler/docs/Riegler-Dissertation.pdf">Simple enumeration formulas related to Alternating Sign Monotone Triangles and standard Young tableaux</a>, Dissertation, Universitat Wien, 2014.

%H D. P. Robbins, <a href="http://dx.doi.org/10.1007/BF03024081">The story of 1, 2, 7, 42, 429, 7436, ...</a>, Math. Intellig., 13 (No. 2, 1991), 12-19.

%H D. P. Robbins, <a href="https://arxiv.org/abs/math/0008045">Symmetry classes of alternating sign matrices</a>, arXiv:math/0008045 [math.CO], 2000.

%H R. P. Stanley, <a href="http://dx.doi.org/10.1007/BFb0072521">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

%H R. P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

%H Yu. G. Stroganov, <a href="http://arXiv.org/abs/math-ph/0304004">3-enumerated alternating sign matrices</a>

%H X. Sun, V. H. Moll, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Moll/moll.html">The p-adic Valuations of Sequences Counting Alternating Sign Matrices</a>, JIS 12 (2009) 09.3.8

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingSignMatrix.html">Alternating Sign Matrix</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DescendingPlanePartition.html">Descending Plane Partition</a>

%H D. Zeilberger, <a href="https://arxiv.org/abs/math/9407211">Proof of the alternating-sign matrix conjecture</a>, arXiv:math/9407211 [math.CO], 1994.

%H D. Zeilberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r13">Proof of the alternating-sign matrix conjecture</a>, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.

%H D. Zeilberger, <a href="https://arxiv.org/abs/math/9606224">Proof of the Refined Alternating Sign Matrix Conjecture</a>, arXiv:math/9606224 [math.CO], 1996.

%H D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/amrr.html">A constant term identity featuring the ubiquitous(and mysterious)Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,...</a>, J. Combin. Theory, A 66 (1994), 17-27.

%H D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/DaveRobbins/guess.html">Dave Robbins's Art of Guessing</a>, Adv. in Appl. Math. 34 (2005), 939-954.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.

%F The Hankel transform of A025748 is a(n) * 3^binomial(n, 2). - _Michael Somos_, Aug 30 2003

%F a(n) = sqrt(A049503).

%F From _Bill Gosper_, Mar 11 2014: (Start)

%F A "Stirling's formula" for this sequence is

%F a(n) ~ 3^(5/36+(3/2)*n^2)/(2^(1/4+2*n^2)*n^(5/36))*(exp(zeta'(-1))*gamma(2/3)^2/Pi)^(1/3).

%F which gives results which are very close to the true values:

%F 1.0063254118710128, 2.003523267231662,

%F 7.0056223910285915, 42.01915917750558,

%F 429.12582410098327, 7437.518404899576,

%F 218380.8077275304, 1.085146545456063*^7,

%F 9.119184824937415*^8

%F (End)

%F a(n+1) = a(n) * n! * (3*n+1)! / ((2*n)! * (2*n+1)!). - _Reinhard Zumkeller_, Sep 30 2014; corrected by _Eric W. Weisstein_, Nov 08 2016

%e G.f. = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 429*x^5 + 7436*x^6 + 218348*x^7 + ...

%p A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;

%p # _Bill Gosper_'s approximation (for n>0):

%p a_prox := n -> (2^(5/12-2*n^2)*3^(-7/36+1/2*(3*n^2))*exp(1/3*Zeta(1,-1))*Pi^(1/3)) /(n^(5/36)*GAMMA(1/3)^(2/3)); # _Peter Luschny_, Aug 14 2014

%t f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 0, 17}] (* _Robert G. Wilson v_, Jul 15 2004 *)

%t a[ n_] := If[ n < 0, 0, Product[(3 k + 1)! / (n + k)!, {k, 0, n - 1}]]; (* _Michael Somos_, May 06 2015 *)

%o (PARI) {a(n) = if( n<0, 0, prod(k=0, n-1, (3*k + 1)! / (n + k)!))}; /* _Michael Somos_, Aug 30 2003 */

%o (PARI) {a(n) = my(A); if( n<0, 0, A = Vec( (1 - (1 - 9*x + O(x^(2*n)))^(1/3)) / (3*x)); matdet( matrix(n, n, i, j, A[i+j-1])) / 3^binomial(n,2))}; /* _Michael Somos_, Aug 30 2003 */

%Y Cf. A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A160707, A160708, A194827, A227833.

%K nonn,easy,nice,core

%O 0,3

%A _N. J. A. Sloane_

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Last modified September 23 06:51 EDT 2018. Contains 315273 sequences. (Running on oeis4.)