A352656 -id:A352656 - cp's OEIS frontend

A008793 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.

1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000

Offset: 0

Jonas Wallgren

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

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).

Seiichi Manyama, Table of n, a(n) for n = 0..54 (terms 0..30 from T. D. Noe)
T. Amdeberhan and V. 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).
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.
I. Fischer, Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center, arXiv:math/9906102 [math.CO], 1999.
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
M. Fulmek and C. 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.
I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, Topological properties of circumcoronenes, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - _Emeric Deutsch_, May 14 2018
H. Helfgott and I. 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.
C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.
P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University 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
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.

Cf. A000984, A066931, A352656, A352657. Main diagonal of array A103905.

A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;

Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]

a(n) = prod(i=1,n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ Michel Marcus, Jul 13 2020

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

More terms from Eric W. Weisstein

A352657 The number of lozenge tilings of a semiregular hexagon of side lengths n, n, 3n, n, n and 3n; equivalently, the number of plane partitions whose solid Young diagram fits inside an n X n X 3*n box.

Original entry on oeis.org

1, 4, 336, 572572, 19571505408, 13365232267026024, 182001937855822420050000, 49372092168218024268166702560000, 266640931683989945767062736068603511111680, 28657545169614835585678719963104037818950931553412096, 61277278161726929232430881966673334396569563602790616552072890176

Offset: 0

Peter Bala, Apr 22 2022

A lozenge is a unit rhombus with internal angles of 60 and 120 degrees. A hexagon is semiregular if its internal angles are 120 degrees and opposite sides are of equal length. Let S(n) = Product_{k = 0..n-1} k! = A000178(n-1) for n >= 1. S(n) equals the superfactorial of n-1. Then for a, b and c nonnegative integers a semiregular hexagon with side-lengths a, b, c, a, b, c can be tiled by lozenges in exactly S(a+b+c)*S(a)*S(b)*S(c)/(S(a+b)*S(a+c)*S(b+c)) ways.

The superfactorial ratio (S(a)*S(b)*S(c)*S(a+b+c))/(S(a+b)*S(a+c)*S(b+c)) is an integer (see MacMahon, Chapter II, Section 429, p. 182, with x -> 1) and can be viewed as the superfactorial analog of the binomial coefficient (a + b)!/(a!*b!). Setting a = b = n, c = 3*n gives the entries for the present sequence, a superfactorial analog of A005810(n) = binomial(4*n,n).

			Examples of supercongruences:
p = 5, n = 1, r = 1:
a(5) - a(1)^5 = 13365232267026024 - 4^5 = (2^3)*(5^5)*534609290681 == 0 (mod 5^5).
p = 7, n = 1, r = 1:
a(7) - a(1)^7 = 49372092168218024268166702560000 - 4^7 = (2^8)*(7^4)*42153329 *1905537621534581059 == 0 (mod 7^4).
p = 3, n = 1, r = 2:
a(3^2) - a(3)^3 = 28657545169614835585678719963104037818950931553412096 - 572572^3 = (2^6)*(3^9)*7*13*36206433373771931*6904632711001213215426713099 == 0 (mod 3^9).
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 4*x + 176*x^2 + 191540*x^3 + 4893655248*x^4 + 2673066058559752*x^5 + 30333667002369040991520*x^6 + 7053156145366242954671905412736*x^7 + 33330116488711372656254906993570075436704*x^8 + 3184171685646079976603214029980784880572652377971904*x^9 + 6127727816185429609991005336553574169498938182021433716145181760*x^10 + ....

C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.
P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Superfactorial

Cf. A000178, A005810, A008793, A074962, A352656.

S := proc(n) local i; mul(i!, i = 0..n-1) end proc:
a := n -> S(n)^2*S(3*n)*S(5*n)/(S(2*n)*S(4*n)^2):
seq(a(n), n = 0..10);

Table[BarnesG[n + 1]^2 * BarnesG[3*n + 1] * BarnesG[5*n + 1] / (BarnesG[2*n + 1] * BarnesG[4*n + 1]^2), {n, 0, 10}] (* Vaclav Kotesovec, May 16 2022 *)

a(n) = S(n)^2*S(3*n)*S(5*n)/(S(2*n)*S(4*n)^2), where S(n) = Product_{k = 0..n-1} k! with S(0) = 1.
a(n) = Product_{i = 1..3*n} (2*n+i-1)!*(i-1)!/(n+i-1)!^2.
a(n) = Product_{i = 1..n} (4*n+i-1)!*(i-1)!/((3*n+i-1)!*(n+i-1)!).
a(n) = Product_{i = 1..3*n} Product_{1 <= j, k <= n} (i + j + k - 1)/(i + j + k - 2).
a(n) = Product_{i = 1..n} Product_{j = 1..n} (3*n + i + j - 1)/(i + j - 1).
a(n) = Product_{i = 1..3*n} Product_{j = 1..n} (n + i + j - 1)/(i + j - 1).
For n >= 1, a(n) = det( (binomial(4*n,n+i-j)) ) for 1 <= i, j <= n. Apply Krattenhaller, Theorem 4 with a = n, b = 3*n and c = n.
a(n) ~ 1/A*(32/(15*n))^(1/12)*exp(B*n^2 + 1/12), where A = 1.2824271291... is the Glaisher-Kinkelin constant A074962 and B = (25/2)*log(5) + (9/2)*log(3) - 34*log(2).
Conjecture 1): the Gauss congruences a(n*p^r) == a(n*p^(r-1)) (mod p^r) hold for all primes p and positive integers n and r. If true, then the expansion of exp(Sum_{n >= 1} a(n)*x^n/n) has integer coefficients.
Conjecture 2): 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.
a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) * 5^(25*n^2/2 - 1/12) / (A * n^(1/12) * 2^(34*n^2 - 5/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 16 2022
From Peter Bala, Feb 15 2023: (Start)
a(n) = Product_{i = 1..n} Product_{j = 3*n..4*n-1} (i+j) / Product_{j = 0..n-1} (i+j).
a(n) = Product_{i = 1..3*n} Product_{j = n..2*n-1} (i+j) / Product_{j = 0..n-1} (i+j). (End)

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A008793 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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A352657 The number of lozenge tilings of a semiregular hexagon of side lengths n, n, 3n, n, n and 3n; equivalently, the number of plane partitions whose solid Young diagram fits inside an n X n X 3*n box.

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cp's OEIS Frontend

A008793 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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Keywords

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A352657 The number of lozenge tilings of a semiregular hexagon of side lengths n, n, 3*n, n, n and 3*n; equivalently, the number of plane partitions whose solid Young diagram fits inside an n X n X 3*n box.

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Author

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Examples

Links

Crossrefs

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Maple

Mathematica

Formula

A352657 The number of lozenge tilings of a semiregular hexagon of side lengths n, n, 3n, n, n and 3n; equivalently, the number of plane partitions whose solid Young diagram fits inside an n X n X 3*n box.