%I A005130 M1808
%S A005130 1,1,2,7,42,429,7436,218348,10850216,911835460,129534272700,31095744852375,
%T A005130 12611311859677500,8639383518297652500,9995541355448167482000,19529076234661277104897200,
%U A005130 64427185703425689356896743840,358869201916137601447486156417296
%N 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).
%C A005130 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 A005130 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 27 2009: 
               (Start)
%C A005130 Starting with offset 1 = row sums of triangle A160708, and convolution 
               square of A160707.
%C A005130 a(n) is odd iff n is a Jacobsthal number [Frey and Sellers, 2000].
%C A005130 Starting with offset 1 = row sums of triangle A160708.
%C A005130 Starting (1, 2, 7,...) = convolution square of A160707: [1, 1, 3, 18, 
               192,...]. (End)
%D A005130 G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, 
               Invent. Math., 53 (1979), 193-225. (See Theorem 10.)
%D A005130 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n 
               on page 4, D_r on page 197.
%D A005130 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture 
               was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
%D A005130 M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary 
               and totally symmetric, self-complementary plane partitions, J. Combin. 
               Theory Ser. A 86 (1999), 382-389.
%D A005130 C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, 
               Oxford Univ. Press NY 2001.
%D A005130 D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 
               13 (No. 2, 1991), 12-19.
%D A005130 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005130 R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, 
               pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. 
               Notes Math. 1234, 1986.
%D A005130 D. Zeilberger, A constant term identity featuring the ubiquitous (and 
               mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, 
               ..., J. Combin. Theory, A 66 (1994), 17-27.
%D A005130 D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 
               (2005), 939-954.
%H A005130 T. D. Noe, <a href="b005130.txt">Table of n, a(n) for n = 0..100</a>
%H A005130 M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="http://arXiv.org/
               abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/
               2, alternating sign matrices and plane partitions, arXiv cond-mat/
               0101385</a>
%H A005130 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 A005130 F. Colomo and A. G. Pronko, <a href="http://it.arXiv.org/abs/math-ph/
               0411076">Square ice, alternating sign matrices and classical orthogonal 
               polynomials</a>, JSTAT (2005) P01005.
%H A005130 I. Fischer, <a href="http://arXiv.org/abs/math.CO/0501102">The number 
               of monotone triangles with prescribed bottom row</a>
%H A005130 P. Di Francesco, <a href="http://arXiv.org/abs/cond-mat/0409576">A refined 
               Razumov-Stroganov conjecture II</a>
%H A005130 P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="http://arXiv.org/
               abs/math-ph/0410002">Determinant formulae for some tiling problems...</
               a>
%H A005130 D. D. Frey and J. A. Sellers, Journal of Integer Sequences Vol. 3 (2000) 
               #00.2.3, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/
               sellers.pdf">Jacobsthal Numbers and Alternating Sign Matrices</a>
%H A005130 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 A005130 J. de Gier, <a href="http://arXiv.org/abs/math.CO/0211285">Loops, matchings 
               and alternating-sign matrices</a>
%H A005130 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 A005130 G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/0008184">Symmetry 
               classes of alternating-sign matrices under one roof, arXiv math.CO/
               0008184</a>
%H A005130 J. Propp, <a href="http://www.dmtcs.org/pdfpapers/dmAA0103.pdf">The many 
               faces of alternating-sign matrices.</a>
%H A005130 A. V. Razumov and Yu. G. Stroganov, <a href="http://arXiv.org/abs/cond-mat/
               0012141">Spin chains and combinatorics, arXiv cond-mat/0012141</a>
%H A005130 D. P. Robbins, Symmetry classes of alternating sign matrices, <a href="http:/
               /arXiv.org/abs/math.CO/0008045">arXiv:math.CO/0008045</a>
%H A005130 Yu. G. Stroganov, <a href="http://arXiv.org/abs/math-ph/0304004">3-enumerated 
               alternating sign matrices</a>
%H A005130 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingSignMatrix.html">Link to a section of The World of Mathematics 
               (1).</a>
%H A005130 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DescendingPlanePartition.html">Link to a section of The World of 
               Mathematics (2).</a>
%H A005130 D. Zeilberger, Proof of the alternating-sign matrix conjecture, <a href="http:/
               /arXiv.org/abs/math.CO/9407211">arXiv:math.CO/9407211</a>
%H A005130 D. Zeilberger, Proof of the alternating-sign matrix conjecture, <a href="http:/
               /www.combinatorics.org/Volume_3/volume3_2.html#R13">Elec. J. Combin., 
               Vol. 3 (Number 2) (1996), #R13</a>.
%H A005130 D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9606224">[math/9606224] 
               Proof of the Refined Alternating Sign Matrix Conjecture</a>
%H A005130 D. Zeilberger, <a href="http://www.math.temple.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>
%H A005130 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 A005130 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%H A005130 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A005130 a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.
%F A005130 The Hankel transform of A025748 is a(n)3^binomial(n,2).
%F A005130 a(n) = sqrt(A049503).
%p A005130 A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;
%t A005130 f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 
               0, 17}] (from Robert G. Wilson v Jul 15 2004)
%o A005130 (PARI) a(n)=if(n<0,0,prod(k=0,n-1,(3*k+1)!/(n+k)!))
%o A005130 (PARI) a(n)=local(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))
%Y A005130 Cf. A006366, A048601, also A003827, A005156, A005158, A005160-A005164, 
               A050204, A049503.
%Y A005130 Cf. A160707, A160708.
%Y A005130 Sequence in context: A066383 A011802 A007065 this_sequence A091669 A108042 
               A152559
%Y A005130 Adjacent sequences: A005127 A005128 A005129 this_sequence A005131 A005132 
               A005133
%K A005130 nonn,easy,nice,core
%O A005130 0,3
%A A005130 N. J. A. Sloane (njas(AT)research.att.com).