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%I #28 Jan 10 2019 08:09:01
%S 1,1,2,1,6,3,1,20,20,4,1,70,175,50,5,1,252,1764,980,105,6,1,924,19404,
%T 24696,4116,196,7,1,3432,226512,731808,232848,14112,336,8,1,12870,
%U 2760615,24293412,16818516,1646568,41580,540,9,1,48620,34763300
%N Square array T(n,k) read by antidiagonals: number of tilings of an <n,k,n> hexagon.
%C As a square array, T(n,k) = number of all k-watermelons without a wall of length n. - _Steven Finch_, Mar 30 2008
%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 A. J. Guttmann, A. L. Owczarek and X. G. Viennot, <a href="https://doi.org/10.1088/0305-4470/31/40/007">Vicious walkers and Young tableaux. I. Without walls</a>, J. Phys. A 31 (1998) 8123-8135.
%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 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/0503507 [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.
%F 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).
%F T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
%F T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].
%F 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
%F 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
%F For k >= 1, T(n,k)=det(binomial(2*n,n+i-j))1<=i,j<=k [Krattenhaller, Theorem 4].
%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, Section 4.29 with x -> 1]. Setting a = b = n and c = k gives the entries for this table. - Peter Bala, Dec 22 2011
%e Array begins:
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 6, 20, 50, 105, 196, ...
%e 1, 20, 175, 980, 4116, 14112, ...
%e 1, 70, 1764, 24696, 232848, 1646568, ...
%e 1, 252, 19404, 731808, 16818516, 267227532, ...
%e ...
%t 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 *)
%Y Rows include A002415, A047819, A047835, A047831.
%Y Columns include A000984 and A000891.
%Y Main diagonal is A008793.
%Y Cf. A120258, A133112.
%K nonn,tabl
%O 1,3
%A _Ralf Stephan_, Feb 22 2005
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