A103905 Square array T(n,k) read by antidiagonals: number of tilings of an <n,k,n> hexagon.

1, 1, 2, 1, 6, 3, 1, 20, 20, 4, 1, 70, 175, 50, 5, 1, 252, 1764, 980, 105, 6, 1, 924, 19404, 24696, 4116, 196, 7, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1, 48620, 34763300

Offset

1,3

Comments

As a square array, T(n,k) = number of all k-watermelons without a wall of length n. - Steven Finch, Mar 30 2008

Links

Table of n, a(n) for n=1..48.

P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.

A. J. Guttmann, A. L. Owczarek and X. G. Viennot, Vicious walkers and Young tableaux. I. Without walls, J. Phys. A 31 (1998) 8123-8135.

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.

C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2005.

P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.

Formula

T(n, k) = [V(2n+k-1)V(k-1)V(n-1)^2]/[V(2n-1)V(n+k-1)^2], with V(n) the superfactorial numbers (A000178).

T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].

T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].

T(n, k) = Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul Barry, Jun 13 2006

Conjectural formula as a sum of squares of Vandermonde determinants: T(n,k) = 1/((1!*2! ... *(n-1)!)^2*n!)* sum {1 <= x_1, ..., x_n <= k} (det V(x_1, ...,x_n))^2, where V(x_1, ...,x_n} is the Vandermonde matrix of order n. Compare with A133112. - Peter Bala, Sep 18 2007

For k >= 1, T(n,k)=det(binomial(2*n,n+i-j))1<=i,j<=k [Krattenhaller, Theorem 4].

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, Section 4.29 with x -> 1]. Setting a = b = n and c = k gives the entries for this table. - Peter Bala, Dec 22 2011

Example

Array begins:
  1,   2,     3,      4,        5,         6, ...
  1,   6,    20,     50,      105,       196, ...
  1,  20,   175,    980,     4116,     14112, ...
  1,  70,  1764,  24696,   232848,   1646568, ...
  1, 252, 19404, 731808, 16818516, 267227532, ...
  ...

Mathematica

t[n_, k_] := Product[j!*(j + 2*n)!/(j + n)!^2, {j, 0, k - 1}]; Join[{1}, Flatten[ Table[ t[n - k , k], {n, 1, 10}, {k, 1, n}]]] (* Jean-François Alcover, May 16 2012, from 2nd formula *)

Crossrefs

Rows include A002415, A047819, A047835, A047831.

Columns include A000984 and A000891.

Main diagonal is A008793.

Cf. A120258, A133112.

Sequence in context: A175757 A060539 A163269 * A270967 A103209 A089900

Adjacent sequences: A103902 A103903 A103904 * A103906 A103907 A103908

Keyword

nonn,tabl

Author

Ralf Stephan, Feb 22 2005

Status

approved