%I A103905
%S A103905 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 A103905 24696,4116,196,7,1,3432,226512,731808,232848,14112,336,8,1,12870,
%U A103905 2760615,24293412,16818516,1646568,41580,540,9,1,48620,34763300
%N A103905 Square array T(n,k) read by antidiagonals: number of tilings of an <n,
               k,n> hexagon.
%C A103905 As a square array, T(n,k) = number of all k-watermelons without a wall 
               of length n. - S. R. Finch (Steven.Finch(AT)inria.fr), Mar 30 2008
%D A103905 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 A103905 P. J. Forrester and A. Gamburd, <a href="http://arXiv.org/abs/math.CO/
               0503002">Counting formulas associated with some random matrix averages</
               a>
%H A103905 H. Helfgott and I. M. Gessel, <a href="http://arXiv.org/abs/math.CO/9810143">
               Enumeration of tilings of diamonds and hexagons with defects</a>
%F A103905 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 A103905 T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
%F A103905 T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) 
               ]]].
%F A103905 T(n,k)=Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul 
               Barry (pbarry(AT)wit.ie), Jun 13 2006
%F A103905 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 (pbala(AT)toucansurf.com), 
               Sep 18 2007
%e A103905 Array begins:
%e A103905 1,2,3,4,5,6,
%e A103905 1,6,20,50,105,196,
%e A103905 1,20,175,980,4116,14112,
%e A103905 1,70,1764,24696,232848,1646568,
%e A103905 1,252,19404,731808,16818516,267227532,
%Y A103905 Rows include A002415, A047819, A047835, A047831. Columns include A000984 
               and A000891. Main diagonal is A008793.
%Y A103905 Cf. A133112.
%Y A103905 Sequence in context: A128741 A060539 A163269 this_sequence A103209 A089900 
               A138533
%Y A103905 Adjacent sequences: A103902 A103903 A103904 this_sequence A103906 A103907 
               A103908
%K A103905 nonn,tabl
%O A103905 1,3
%A A103905 Ralf Stephan, Feb 22 2005