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 A103905 Square array T(n,k) read by antidiagonals: number of tilings of an hexagon. 12
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)