A005157 -id:A005157 - 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

A049505 a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.

Original entry on oeis.org

1, 2, 10, 112, 2772, 151008, 18076916, 4751252480, 2740612658576, 3468301123758080, 9627912669442441500, 58618653300361405440000, 782683432110638830001250000, 22916694891747599820616089600000, 1471328419282772010324439370939640000

Offset: 0

N. J. A. Sloane

The first printing of the Bressoud book states that the formula Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1) in Eq. (6.8) is the number of totally symmetric plane partitions. This is wrong, although it does produce the current sequence. For the correct formula for the number of totally symmetric plane partitions see A005157.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198.

T. D. Noe, Table of n, a(n) for n = 0..40
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
P. J. Taylor, Counting distinct dimer hex tilings, arXiv:1602.06796 [math.CO], 2016.

Main diagonal of array A102539.
Main diagonal of array in A073165.
Cf. A005157, A008793.

A049505 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+j-1),j=i..n),i=1..n); end;

a[n_] := Product[ ((2i-2)!*(i+2n-1)!)/((i+n-1)!*(2i+n-2)!), {i, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 22 2012, after PARI *)

a(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+j-1)))

a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1).
a(n) = Product_{i=1..n} (((2*i-2)!*(i+2*n-1)!)/((i+n-1)!*(2*i+n-2)!)). - Jean-François Alcover, Jun 22 2012
a(n) = Product_{i=1..n} (binomial((i-1) + 2*n, n)/binomial(n + 2*(i-1), n)). - Olivier Gérard, Feb 25 2015
a(n) ~ exp(1/24) * 3^(9*n^2/4 + 3*n/4 - 1/24) / (A^(1/2) * n^(1/24) * 2^(3*n^2 + n/2 + 1/8)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015
From Peter Bala, Feb 15 2023: (Start)
a(n+1) = m(n)*a(n) where m(n) = ((3*n + 2)!*n!^2)/((2 n)!*(2 n + 1)!^2) * Product_{i = 1..n} n + 2*i for n >= 1.
Conjectures:
1) the supercongruence a(p) == 2^((p+1)/2) (mod p^3) holds for all primes p >= 3 (checked up to p = 1009).
2) the congruence a(p^2) == (-1)^((p^2-1)/8)*a(p)^(p^2-p+1) (mod p^3) holds for all primes p >= 3 (checked up to p = 89).
3) the congruence a(p^3) == a(p^2)^((p^3-p^2+2)/2) (mod p^3) holds for all primes p >= 2. (End)

Edited by N. J. A. Sloane, Jun 30 2013; codes and formula checked by N. J. A. Sloane and Olivier Gérard

A184173 Triangle read by rows: T(n,k) is the sum of the k X k minors in the n X n Pascal matrix (0<=k<=n; the empty 0 X 0 minor is defined to be 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 34, 15, 1, 1, 31, 144, 144, 31, 1, 1, 63, 574, 1155, 574, 63, 1, 1, 127, 2226, 8526, 8526, 2226, 127, 1, 1, 255, 8533, 60588, 113832, 60588, 8533, 255, 1, 1, 511, 32587, 424117, 1444608, 1444608, 424117, 32587, 511, 1

Offset: 0

Emeric Deutsch, Jan 12 2011

Apparently, the sum of the entries in row n is A005157(n).

			T(3,1) = 7 because in the 3 X 3 Pascal matrix [1,0,0/1,1,0/1,2,1] the sum of the entries is 7.
Triangle starts:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,    1;
  1,  15,   34,   15,    1;
  1,  31,  144,  144,   31,    1;
  1,  63,  574, 1155,  574,   63,   1;
  1, 127, 2226, 8526, 8526, 2226, 127, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Wikipedia, Minor (linear algebra)

Columns k=0-2 give: A000012, A000225, A306376.

Cf. A005157, A007318.

with(combinat): with(LinearAlgebra):
T:= proc(n, k) option remember; `if`(n-k add(add(
      Determinant(SubMatrix(Matrix(n, (i, j)-> binomial(i-1, j-1)),
       i, j)), j in l), i in l))(choose([$1..n], k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..7);  # Alois P. Heinz, Feb 11 2019

T[n_, k_] := T[n, k] = If[k == 0 || k == n, 1, Module[{l, M},
    l = Subsets[Range[n], {k}];
    M = Table[Binomial[i-1, j-1], {i, n}, {j, n}];
    Total[Det /@ Flatten[Table[M[[i, j]], {i, l}, {j, l}], 1]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019 updated Feb 29 2024 *)

The triangle is symmetric: T(n,k) = T(n,n-k).

Typo corrected by Alois P. Heinz, Feb 11 2019

A323848 Irregular triangle read by rows: T(n,d) (n >= 1, d <= n-1 for n>1) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1), and M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

Original entry on oeis.org

0, 4, 18, 25, 68, 386, 256, 250, 4657, 12200, 4356, 922, 54219, 432842, 608993, 123904, 3430, 642815, 14697256, 60650883, 49489706, 5909761, 12868, 7852836, 514608568, 5713126349, 13458882036, 6648891794, 473497600, 48618, 98755951, 18971384148, 558848240787, 3406380649146, 4857082197177, 1489334202216, 63799687396

Offset: 1

N. J. A. Sloane, Feb 07 2019

T(n,n-1) = A005157(n-1)^2 for n >= 2. See Knuth (2019) link.

			Triangle begins:
  n\d    1      2        3        4        5       6  7
   1     0      0        0        0        0       0  0
   2     4      0        0        0        0       0  0
   3    18     25        0        0        0       0  0
   4    68    386      256        0        0       0  0
   5   250   4657    12200     4356        0       0  0
   6   922  54219   432842   608993   123904       0  0
   7  3430 642815 14697256 60650883 49489706 5909761  0
...

D. E. Knuth, Email to N. J. A. Sloane, Feb 06 2019.

Alois P. Heinz, Rows n = 1..14, flattened
Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.

Columns d=1-2 give: A115112, A306322.

Cf. A005157, A323849.

T(n,1) = binomial(2n,n) - 2.

More terms from Alois P. Heinz, Feb 07 2019

A214564 Number T(n,k) of totally symmetric plane partitions with largest part <= n and exactly k orbits under action of the symmetric group S_3; triangle T(n,k), n>=0, 0<=k<=A000292(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 19, 20, 20, 20, 20, 19, 18, 17, 15, 13, 12, 10, 8, 7, 5, 4, 3, 2, 1, 1, 1

Offset: 0

Alois P. Heinz, Jul 21 2012

			Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1, 1, 1, 1;
  1, 1, 1, 2, 2, 2, 2, 2, 1,  1,  1;
  1, 1, 1, 2, 3, 3, 4, 5, 5,  5,  6,  5,  5,  5,  4,  3,  3,  2,  1,  1,  1;
  1, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 17, 18, 19, 20, 20, 20, 20, 19, ...
  ...

Alois P. Heinz, Rows n = 0..21
C. Koutschan, M. Kauers, and D. Zeilberger, Proof of George Andrews’s and David Robbins’s q-TSPP conjecture, PNAS (2011), 108: 2196-2199.
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.

Row sums give: A005157.

Cf. A000292.

gf:= n-> simplify(mul(mul(mul( (1-q^(i+j+k-1))/
         (1-q^(i+j+k-2)), i=1..j), j=1..k), k=1..n)):
T:= n-> seq(coeff(gf(n), q, k), k=0..n*(n+1)*(n+2)/6):
seq(T(n), n=0..7);

G.f. of row n: Product_{1<=i<=j<=k<=n} (1-q^(i+j+k-1))/(1-q^(i+j+k-2)).

A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

Original entry on oeis.org

1, 3, 15, 126, 1782, 42471, 1706562, 115640460, 13216815036, 2548124192970, 828751754742975, 454739496669274500, 420972227408592675000, 657522745057190417409000, 1732789066323343611643088400, 7704900186426840030325195822560, 57807195523790513335568376591463776

Offset: 0

Michel Marcus, Dec 21 2015

a(n) gives the number of diagonally and antidiagonally symmetric alternating sign matrices (DASASM's) of order (2n+1) X (2n+1) (see Behrend et al. link).

Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka, Diagonally and antidiagonally symmetric alternating sign matrices of odd order, arXiv:1512.06030 [math.CO], 2015.

Cf. A005157, A086205.

[&*[Factorial(3*k)/Factorial(n+k): k in [0..n]]: n in [0..16]]; // Vincenzo Librandi, Dec 21 2015

Table[Product[(3 k)!/(n + k)!, {k, 0, n}], {n, 0, 16}] (* Vincenzo Librandi, Dec 21 2015 *)

PARI
```
a(n) = prod(k=0, n, (3*k)!/(n+k)!);
```

a(n) ~ Gamma(1/3)^(1/3) * exp(1/36) * n^(1/36) * 3^(3*n^2/2 + 2*n + 11/36) / (A^(1/3) * Pi^(1/6) * 2^(2*n^2 + 2*n + 7/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 21 2015

a(n) = Product_{1 <= i <= j <= n} (i + 2*j)/(i + j - 1). Note that Product_{1 <= i <= j <= n} (i + j)/(i + j - 1) = 2^n. - Peter Bala, Feb 19 2023

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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A049505 a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.

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A184173 Triangle read by rows: T(n,k) is the sum of the k X k minors in the n X n Pascal matrix (0<=k<=n; the empty 0 X 0 minor is defined to be 1).

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A323848 Irregular triangle read by rows: T(n,d) (n >= 1, d <= n-1 for n>1) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1), and M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

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A214564 Number T(n,k) of totally symmetric plane partitions with largest part <= n and exactly k orbits under action of the symmetric group S_3; triangle T(n,k), n>=0, 0<=k<=A000292(n), read by rows.

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A266091 a(n) = Product_{k=0..n} (3*k)!/(n+k)!.

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Magma

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A236691 Number of totally symmetric solid partitions which fit in an n X n X n X n box.

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