A003827 -id:A003827 - cp's OEIS frontend

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).

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

Offset: 0

N. J. A. Sloane

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.
a(n) is odd iff n is a Jacobsthal number (A001045) [Frey and Sellers, 2000]. - Gary W. Adamson, May 27 2009

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

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 71, 557, 573.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386.
C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
T. Amdeberhan and 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.)
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, A Riordan array family for some integrable lattice models, arXiv:2409.09547 [math.CO], 2024.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 7.
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 [cond-mat.stat-mech], 2001.
Andrew Beveridge, Ian Calaway, and Kristin Heysse, de Finetti Lattices and Magog Triangles, arXiv:1912.12319 [math.CO], 2019.
E. Beyerstedt, V. H. Moll, and X. Sun, The p-adic Valuation of the ASM Numbers, J. Int. Seq. 14 (2011) # 11.8.7.
Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.
Sara C. Billey, Brendon Rhoades, and Vasu Tewari, Boolean product polynomials, Schur positivity, and Chern plethysm, arXiv:1902.11165 [math.CO], 2019.
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, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
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, arXiv:math-ph/0404045, 2004; Advances in Applied Mathematics 34 (2005) 798.
F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
G. Conant, Magmas and Magog Triangles, 2014.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
P. Di Francesco, Twenty Vertex model and domino tilings of the Aztec triangle, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
P. Di Francesco, P. Zinn-Justin, and J.-B. Zuber, Determinant formulas for some tiling problems..., arXiv:math-ph/0410002, 2004.
FindStat - Combinatorial Statistic Finder, Alternating sign matrices
I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
Ilse Fischer and Manjil P. Saikia, Refined Enumeration of Symmetry Classes of Alternating Sign Matrices, arXiv:1906.07723 [math.CO], 2019.
Ilse Fischer and Matjaz Konvalinka, A bijective proof of the ASM theorem, Part I: the operator formula, arXiv:1910.04198 [math.CO], 2019.
T. Fonseca and F. Balogh, The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials, Journal of Algebraic Combinatorics, 2014, arXiv:1210.4527
D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, Journal of Integer Sequences Vol. 3 (2000) #00.2.3.
D. D. Frey and J. A. Sellers, Prime Power Divisors of the Number of n X n Alternating Sign Matrices
Markus Fulmek, A statistics-respecting bijection between permutation matrices and descending plane partitions without special parts, Electronic journal of combinatorics, 27(1) (2020), #P1.391.
M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
C. Heuberger and 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.
Dylan Heuer, 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 arXiv:1611.03387.
Frederick Huang, The 20 Vertex Model and Related Domino Tilings, Ph. D. Dissertation, UC Berkeley, 2023. See p. 1.
Hassan Isanloo, The volume and Ehrhart polynomial of the alternating sign matrix polytope, Cardiff University (Wales, UK 2019).
Masato Kobayashi, Weighted counting of inversions on alternating sign matrices, arXiv:1904.02265 [math.CO], 2019.
G. Kuperberg, Another proof of the alternating-sign matrix conjecture, arXiv:math/9712207 [math.CO], 1997; Internat. Math. Res. Notices, No. 3, (1996), 139-150.
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001; Ann. Math. 156 (3) (2002) 835-866
W. H. Mills, David P Robbins, and Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013).
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386. [Annotated scanned copy]
J. Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.
A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, arXiv:cond-mat/0012141 [cond-mat.stat-mech], 2000.
Lukas Riegler, Simple enumeration formulas related to Alternating Sign Monotone Triangles and standard Young tableaux, Dissertation, Universitat Wien, 2014.
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/0008045 [math.CO], 2000.
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.
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. Preprint. [Annotated scanned copy]
Yu. G. Stroganov, 3-enumerated alternating sign matrices, arXiv:math-ph/0304004, 2003.
X. Sun and V. H. Moll, The p-adic Valuations of Sequences Counting Alternating Sign Matrices, JIS 12 (2009) 09.3.8.
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/9407211 [math.CO], 1994.
D. Zeilberger, Proof of the alternating-sign matrix conjecture, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.
D. Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.
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. The link is to a comment on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954. The link is to a version on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
Paul Zinn-Justin, Integrability and combinatorics, arXiv:2404.13221 [math.CO], 2024. See p. 12.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences

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

a:=List([0..18],n->Product([0..n-1],k->Factorial(3*k+1)/Factorial(n+k)));; Print(a); # Muniru A Asiru, Jan 02 2019

A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;
# Bill 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 *)
a[ n_] := If[ n < 0, 0, Product[(3 k + 1)! / (n + k)!, {k, 0, n - 1}]]; (* Michael Somos, May 06 2015 *)

{a(n) = if( n<0, 0, prod(k=0, n-1, (3*k + 1)! / (n + k)!))}; /* Michael Somos, Aug 30 2003 */

{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 */

from math import prod, factorial
def A005130(n): return prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)) # Chai Wah Wu, Feb 02 2022

a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.
The Hankel transform of A025748 is a(n) * 3^binomial(n, 2). - Michael Somos, Aug 30 2003
a(n) = sqrt(A049503).
From Bill 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; corrected by Eric W. Weisstein, Nov 08 2016
For n>0, a(n) = 3^(n - 1/3) * BarnesG(n+1) * BarnesG(3*n)^(1/3) * Gamma(n)^(1/3) * Gamma(n + 1/3)^(2/3) / (BarnesG(2*n+1) * Gamma(1/3)^(2/3)). - Vaclav Kotesovec, Mar 04 2021

A006366 Number of cyclically symmetric plane partitions in the n-cube; also number of 2n X 2n half-turn symmetric alternating sign matrices divided by number of n X n alternating sign matrices.

Original entry on oeis.org

1, 2, 5, 20, 132, 1452, 26741, 826540, 42939620, 3752922788, 552176360205, 136830327773400, 57125602787130000, 40191587143536420000, 47663133295107416936400, 95288872904963020131203520, 321195665986577042490185260608

Offset: 0

N. J. A. Sloane

In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1529 and M1530.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.7) on page 198, except the formula given is incorrect. It should be as shown here.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.

Vincenzo Librandi, Table of n, a(n) for n = 0..90
G. E. Andrews,Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225.
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
W. F. Lunnon, The Pascal matrix, Fib. Quart. vol. 15 (1977) pp. 201-204.
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. Preprint. [Annotated scanned copy]
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Cf. A005130, also A003827, A005156, A005158, A005160-A005164, A048601, A050204.

A006366 := proc(n) local i, j; mul((3*i - 1)*mul((n + i + j - 1)/(2*i + j - 1), j = i .. n)/(3*i - 2), i = 1 .. n) end;

Table[Product[(3i-1)/(3i-2) Product[(n+i+j-1)/(2i+j-1),{j,i,n}],{i,n}],{n,0,20}] (* Harvey P. Dale, Apr 17 2013 *)

a(n)=prod(i=0,n-1,(3*i+2)*(3*i)!/(n+i)!)

a(n) = Product_{i=1..n} (((3*i-1)/(3*i-2))*Product_{j=i..n} (n+i+j-1)/(2*i+j-1)).

a(n) ~ exp(1/36) * GAMMA(1/3)^(4/3) * n^(7/36) * 3^(3*n^2/2 + 11/36) / (A^(1/3) * Pi^(2/3) * 2^(2*n^2 + 7/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

cp's OEIS Frontend

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).

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A006366 Number of cyclically symmetric plane partitions in the n-cube; also number of 2n X 2n half-turn symmetric alternating sign matrices divided by number of n X n alternating sign matrices.

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A051055 'Connected' alternating sign n X n matrices, i.e., not made from smaller blocks.

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A173312 Partial sums of A005130.

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