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A008793
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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.
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34
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1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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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 [Krattenhaller, 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
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MAPLE
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A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;
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MATHEMATICA
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Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]
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PROG
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(PARI) a(n) = prod(i=1, n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ Michel Marcus, Jul 13 2020
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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