OFFSET
0,2
COMMENTS
The 3-dimensional analog of A000984. - William Entriken, Aug 06 2013
The largest prime factor of a(n) is the largest prime p < 3*n. Its multiplicity is equal to 3*n-p. This can be proved with the formula of Michel Marcus, for example. - Walter Trump, Feb 11 2023
a(n) is also the number of resonance structures of circumcircum...coronene, where circum is repeated n-2 times where a(1) is the number of resonance structures of benzene (see Gutman et al.). - Yuan Yao, Oct 29 2023
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545, also p. 575 line -1 with a=b=c=n.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198. The first printing of Eq. (6.8) is wrong (see A049505 and A005157), but if one changes the limits in the formula (before it is corrected) to {1 <= i <= r, 1 <= j <= r}, one obtains the present sequence. - N. J. A. Sloane, Jun 30 2013
Gordon G. Cash and Jerry Ray Dias, Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues, J. Math. Chem., 30 (2001), 429-444. [See K, p. 442.]
Sebastien Desreux, Martin Matamala, Ivan Rapaport, Eric Remila, Domino tilings and related models: space of configurations of domains with holes, arXiv:math/0302344, 27 Feb 2003
Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, Martin van Hecke1, Topological defects produce exotic mechanics in complex metamaterials, arXiv preprint 1903.07919, 2019 [See Section "Compatible metamaterials with fully antiferromagnetic interactions" - N. J. A. Sloane, Mar 23 2019]
J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see p. 261).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..54 (terms 0..30 from T. D. Noe)
Tewodros Amdeberhan and Victor H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.
Guy David and Carlos Tomei, The Problem of the Calissons, The American Mathematical Monthly, Vol. 96, No. 5 (May, 1989), pp. 429-431 (3 pages).
Philippe Di Francesco, Paul Zinn-Justin and Jean-Bernard Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.
Ilse Fischer, Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center, arXiv:math/9906102 [math.CO], 1999.
Peter J. Forrester and Alex Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
Markus Fulmek and Christian Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II, arXiv:math/9909038 [math.CO], 1999.
Ivan Gutman, Sven J. Cyvina, and Vesna Ivanov-Petrović, Topological properties of circumcoronenes, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - Emeric Deutsch, May 14 2018
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Sam Hopkins and Tri Lai, Plane partitions of shifted double staircase shape, arXiv:2007.05381 [math.CO], 2020. See Table 1 p. 9.
Christian Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Alg. Appl. (2005) Vol. 411, 68-166; arXiv:math/0503507 [math.CO], 2005.
Percy A. MacMahon, Combinatory Analysis, Vol. 2, Cambridge Univ. Press (1916) reprinted by Chelsea, New York (1960).
Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, and Martin van Hecke, Topological defects produce exotic mechanics in complex metamaterials, arXiv:1903.07919 [cond-mat.soft], 2019.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
James Propp, Updated article
James Propp, Tiling Problems, Old and New, Rutgers University Math Colloquium, March 30, 2022.
Martin Ridout, Professor Preece's tredoku tilings, arXiv:2511.12680 [math.CO], 2025. See p. 41, Section 8.3.
N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Walter Trump, Prime factorization of a(n) for 1..950
Eric Weisstein's World of Mathematics, Plane Partition.
FORMULA
a(n) = Product_{i = 0..n-1} (i^(-i)*(n+i)^(2*i-n)*(2*n+i)^(n-i)).
a(n) = Product_{i = 1..n} Product_{j = 0..n-1} (3*n-i-j)/(2*n-i-j).
a(n) = Product_{i = 1..n} Gamma[i]*Gamma[i+2*n]/Gamma[i+n]^2.
a(n) = Product_{i = 0..n-1} i!*(i+2*n)!/(i+n)!^2.
a(n) = Product_{i = 1..n} Product_{j = n..2*n-1} i+j / Product_{j = 0..n-1} i+j. - Paul Barry, Jun 13 2006
For n >= 1, a(n) = det(binomial(2*n, n+i-j) for 1 <= i,j <= n) [Krattenthaler, Theorem 4, with a = b = c = n].
Let H(n) = Product_{k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Chapter II, Section 429, p. 182, with x -> 1]. Setting a = b = c = n gives the entries for this sequence. - Peter Bala, Dec 22 2011
a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) / (A * n^(1/12) * 2^(6*n^2 - 1/4)), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 27 2015
a(n) = Product_{i = 1..n} Product_{j = 1..n} (n+i+j-1)/(i+j-1). - Michel Marcus, Jul 13 2020
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1))^p (mod p^(4*r)) hold for all primes p and positive integers n and r. - Peter Bala, Apr 07 2022
a(n) = det(binomial(2*n + i, n + j) for 0 <= i,j < n). - Vladimir Kruchinin, Sep 27 2025
MAPLE
A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;
MATHEMATICA
Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]
PROG
(PARI) a(n) = prod(i=1, n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ Michel Marcus, Jul 13 2020
(SageMath)
def M(n): return [binomial(2*n + i, n + j) for i in range(n) for j in range(n)]
def a(n): return det(matrix(ZZ, n, n, M(n)))
print([a(n) for n in range(13)]) # Peter Luschny, Sep 27 2025
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Eric W. Weisstein
STATUS
approved