

A005130


Robbins numbers: a(n) = Product_{k=0..n1} (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)


27



1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700, 31095744852375, 12611311859677500, 8639383518297652500, 9995541355448167482000, 19529076234661277104897200, 64427185703425689356896743840, 358869201916137601447486156417296
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OFFSET

0,3


COMMENTS

Also known as the AndrewsMillsRobbinsRumsey 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,...].
(End)


REFERENCES

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), 1219.
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) AndrewsMillsRobbinsRumsey numbers 1, 2, 7, 42, 429, ..., J. Combin. Theory, A 66 (1994), 1727.
D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939954.


LINKS

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), 193225. (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 condmat/0101385
D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637646.
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, selfcomplementary and totally symmetric, selfcomplementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382389.
F. Colomo and A. G. Pronko, On the refined 3enumeration 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 RazumovStroganov conjecture II
P. Di Francesco, P. ZinnJustin 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 alternatingsign matrices, arXiv:math.CO/0211285
C. Heuberger, H. Prodinger, A precise description of the padic valuation of the number of alternating sign matrices, Intl. J. Numb. Th. 7 (1) (2011) 5769
G. Kuperberg, Another proof of the alternatingsign matrix conjecture, Internat. Math. Res. Notices, No. 3, (1996), 139150.
G. Kuperberg, Symmetry classes of alternatingsign 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, 340359. MR0700040 (85b:05013)
J. Propp, The many faces of alternatingsign matrices.
A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, arXiv condmat/0012141
D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 1219.
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. 285293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
Yu. G. Stroganov, 3enumerated 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 alternatingsign matrix conjecture, arXiv:math.CO/9407211
D. Zeilberger, Proof of the alternatingsign 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)AndrewsMillsRobbinsRamsey numbers 1,2,7,42,429,..., J. Combin. Theory, A 66 (1994), 1727.
D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939954.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences


FORMULA

a(n) = Product_{k=0..n1} (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,
9.119184824937415*^8
(End)
a(n+1) = a(n) * n! * (3*n+1)! / ((2*n)! * (2*n+1)).  Reinhard Zumkeller, Sep 30 2014


MAPLE

A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!, k=0..n1); end;
# Gosper's approximation (for n>0):
a_prox := n > (2^(5/122*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


MATHEMATICA

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


PROG

(PARI) a(n)=if(n<0, 0, prod(k=0, n1, (3*k+1)!/(n+k)!))
(PARI) a(n)=local(A); if(n<0, 0, A=Vec((1(19*x+O(x^(2*n)))^(1/3))/(3*x)); matdet(matrix(n, n, i, j, A[i+j1]))/3^binomial(n, 2))


CROSSREFS

Cf. A006366, A048601, also A003827, A005156, A005158, A005160A005164, A050204, A049503, A160707, A160708, A194827, A227833.
Sequence in context: A066383 A011802 A007065 * A091669 A108042 A152559
Adjacent sequences: A005127 A005128 A005129 * A005131 A005132 A005133


KEYWORD

nonn,easy,nice,core


AUTHOR

N. J. A. Sloane.


STATUS

approved



