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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)
1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700, 31095744852375, 12611311859677500, 8639383518297652500, 9995541355448167482000, 19529076234661277104897200, 64427185703425689356896743840, 358869201916137601447486156417296 (list; graph; refs; listen; history; text; internal format)



Also known as the Andrews-Mills-Robbins-Rumsey numbers. - N. J. A. Sloane, May 24 2013

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.

From Gary W. Adamson, May 27 2009: (Start)

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

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

Starting with offset 1 = row sums of triangle A160708.

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



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

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

D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 12-19.

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

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. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.


T. D. Noe, Table of n, a(n) for n = 0..100

T. Amdeberhan, V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.

G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)

M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv cond-mat/0101385

D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.

H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605, 2013

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.

F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, Advances in Applied Mathematics 34 (2005) 798.

F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, JSTAT (2005) P01005.

I. Fischer, The number of monotone triangles with prescribed bottom row

P. Di Francesco, A refined Razumov-Stroganov conjecture II

P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant formulae for some tiling problems...

D. D. Frey and J. A. Sellers, Journal of Integer Sequences Vol. 3 (2000) #00.2.3, Jacobsthal Numbers and Alternating Sign Matrices

D. D. Frey and J. A. Sellers, Prime Power Divisors of the Number of n X n Alternating Sign Matrices

J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math.CO/0211285

C. Heuberger, H. Prodinger, A precise description of the p-adic valuation of the number of alternating sign matrices, Intl. J. Numb. Th. 7 (1) (2011) 57-69

G. Kuperberg, Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Notices, No. 3, (1996), 139-150.

G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184

W. H. Mills, David P Robbins, Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013)

J. Propp, The many faces of alternating-sign matrices.

A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, arXiv cond-mat/0012141

D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 12-19.

D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math.CO/0008045

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.

Yu. G. Stroganov, 3-enumerated alternating sign matrices

Eric Weisstein's World of Mathematics, Alternating Sign Matrix

Eric Weisstein's World of Mathematics, Descending Plane Partition

D. Zeilberger, Proof of the alternating-sign matrix conjecture, arXiv:math.CO/9407211

D. Zeilberger, Proof of the alternating-sign matrix conjecture, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.

D. Zeilberger, [math/9606224] Proof of the Refined Alternating Sign Matrix Conjecture

D. Zeilberger, A constant term identity featuring the ubiquitous(and mysterious)Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,..., J. Combin. Theory, A 66 (1994), 17-27.

D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.

Index entries for sequences related to factorial numbers

Index entries for "core" sequences


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

The Hankel transform of A025748 is a(n)3^binomial(n,2).

a(n) = sqrt(A049503).

From R. W. Gosper, Mar 11, 2014: (Start)

A "Stirling's formula" for this sequence is

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).

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

1.0063254118710128, 2.003523267231662,

7.0056223910285915, 42.01915917750558,

429.12582410098327, 7437.518404899576,

218380.8077275304, 1.085146545456063*^7,



a(n+1) = a(n) * n! * (3*n+1)! / ((2*n)! * (2*n+1)). - Reinhard Zumkeller, Sep 30 2014


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

# Gosper's approximation (for n>0):

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


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


(PARI) a(n)=if(n<0, 0, prod(k=0, n-1, (3*k+1)!/(n+k)!))

(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))


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

Sequence in context: A066383 A011802 A007065 * A091669 A108042 A152559

Adjacent sequences:  A005127 A005128 A005129 * A005131 A005132 A005133




N. J. A. Sloane.



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Last modified December 19 05:11 EST 2014. Contains 252175 sequences.