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%I #128 Dec 22 2023 10:38:47
%S 1,2,20,980,232848,267227532,1478619421136,39405996318420160,
%T 5055160684040254910720,3120344782196754906063540800,
%U 9265037718181937012241727284450000,132307448895406086706107959899799334375000
%N The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.
%C The 3-dimensional analog of A000984. - _William Entriken_, Aug 06 2013
%C 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
%C 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
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545, also p. 575 line -1 with a=b=c=n.
%D 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
%D 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.]
%D 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
%D 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]
%D 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).
%H Seiichi Manyama, <a href="/A008793/b008793.txt">Table of n, a(n) for n = 0..54</a> (terms 0..30 from T. D. Noe)
%H T. Amdeberhan and V. H. Moll, <a href="https://doi.org/10.37236/1997">Arithmetic properties of plane partitions</a>, El. J. Comb. 18 (2) (2011) # P1.
%H Guy David and Carlos Tomei, <a href="https://www.jstor.org/stable/2325150">The Problem of the Calissons</a>, The American Mathematical Monthly, Vol. 96, No. 5 (May, 1989), pp. 429-431 (3 pages).
%H P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/math-ph/0410002">Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops</a>, arXiv:math-ph/0410002, 2004.
%H I. Fischer, <a href="https://arxiv.org/abs/math/9906102">Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center</a>, arXiv:math/9906102 [math.CO], 1999.
%H P. J. Forrester and A. Gamburd, <a href="https://arxiv.org/abs/math/0503002">Counting formulas associated with some random matrix averages</a>, arXiv:math/0503002 [math.CO], 2005.
%H M. Fulmek and C. Krattenthaler, <a href="https://arxiv.org/abs/math/9909038">The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II</a>, arXiv:math/9909038 [math.CO], 1999.
%H I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, <a href="https://doi.org/10.1515/zna-1998-0810">Topological properties of circumcoronenes</a>, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - _Emeric Deutsch_, May 14 2018
%H H. Helfgott and I. M. Gessel, <a href="https://arxiv.org/abs/math/9810143">Enumeration of tilings of diamonds and hexagons with defects</a>, arXiv:math/9810143 [math.CO], 1998.
%H Sam Hopkins and Tri Lai, <a href="https://arxiv.org/abs/2007.05381">Plane partitions of shifted double staircase shape</a>, arXiv:2007.05381 [math.CO], 2020. See Table 1 p. 9.
%H C. Krattenthaler, <a href="https://arxiv.org/abs/math/0503507">Advanced Determinant Calculus: A Complement</a>, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.
%H P. A. MacMahon, <a href="http://www.archive.org/details/combinatoryanaly02macmuoft">Combinatory Analysis, vol. 2</a>, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
%H Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, and Martin van Hecke, <a href="https://arxiv.org/abs/1903.07919">Topological defects produce exotic mechanics in complex metamaterials</a>, arXiv:1903.07919 [cond-mat.soft], 2019.
%H James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H James Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>
%H James Propp, <a href="https://faculty.uml.edu//jpropp/rutgers22.pdf">Tiling Problems, Old and New</a>, Rutgers University Math Colloquium, March 30, 2022
%H N. C. Saldanha and C. Tomei, <a href="https://arxiv.org/abs/math/9801111">An overview of domino and lozenge tilings</a>, arXiv:math/9801111 [math.CO], 1998.
%H P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.
%H Walter Trump, <a href="/A008793/a008793.txt">Prime factorization of a(n) for 1..950</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlanePartition.html">Plane Partition.</a>
%F a(n) = Product_{i = 0..n-1} (i^(-i)*(n+i)^(2*i-n)*(2*n+i)^(n-i)).
%F a(n) = Product_{i = 1..n} Product_{j = 0..n-1} (3*n-i-j)/(2*n-i-j).
%F a(n) = Product_{i = 1..n} Gamma[i]*Gamma[i+2*n]/Gamma[i+n]^2.
%F a(n) = Product_[i = 0..n-1} i!*(i+2*n)!/(i+n)!^2 .
%F 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
%F For n >= 1, a(n) = det(binomial(2*n,n+i-j)) for 1<=i,j<=n [Krattenhaller, Theorem 4, with a = b = c = n].
%F 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
%F 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
%F a(n) = Product_{i = 1..n} Product_{j = 1..n} (n+i+j-1)/(i+j-1). - _Michel Marcus_, Jul 13 2020
%F 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
%p A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;
%t Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]
%o (PARI) a(n) = prod(i=1,n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ _Michel Marcus_, Jul 13 2020
%Y Cf. A000984, A066931, A352656, A352657. Main diagonal of array A103905.
%K nonn,easy,nice
%O 0,2
%A _Jonas Wallgren_
%E More terms from _Eric W. Weisstein_
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