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A103905
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Square array T(n,k) read by antidiagonals: number of tilings of an <n,k,n> hexagon.
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10
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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
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REFERENCES
| 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.
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LINKS
| P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects
C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO].
P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
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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 (pbarry(AT)wit.ie), 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 (pbala(AT)toucansurf.com), 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
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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,
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CROSSREFS
| Rows include A002415, A047819, A047835, A047831. Columns include A000984 and A000891. Main diagonal is A008793.
Cf. A133112.
Sequence in context: A175757 A060539 A163269 * A103209 A089900 A138533
Adjacent sequences: A103902 A103903 A103904 * A103906 A103907 A103908
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KEYWORD
| nonn,tabl
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AUTHOR
| Ralf Stephan, Feb 22 2005
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