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Search: id:A078920
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| A078920 |
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Upper triangle of Catalan Number Wall. |
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+0
4
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| 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 14, 4, 1, 1, 42, 84, 30, 5, 1, 1, 132, 594, 330, 55, 6, 1, 1, 429, 4719, 4719, 1001, 91, 7, 1, 1, 1430, 40898, 81796, 26026, 2548, 140, 8, 1, 1, 4862, 379236, 1643356, 884884, 111384, 5712, 204, 9, 1, 1, 16796, 3711916, 37119160
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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As square array: number of certain symmetric plane partitions, see Forrester/Gamburd paper.
Formatted as a square array, the column k gives the Hankel transform of the Catalan numbers (A000108) beginning at A000108(k) ; example : Hankel transform of [42,132,429,1430,4862, ...]is [42,594,4719,26026,111384, ...](see A091962) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 12 2007
As square array T(n,k): number of all k-watermelons with a wall of length n. - Ralf Stephan, May 09 2007
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LINKS
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M. Somos, Number Walls in Combinatorics.
R. Bacher, Matrices related to the Pascal triangle.
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages
M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall
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FORMULA
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T(n, k) = Prod[i=1..n-k, Prod[j=i..n-k, (i+j+2n)/(i+j) ]].
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EXAMPLE
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1; 1,1; 1,2,1; 1,5,3,1; 1,14,14,4,1; ...
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PROGRAM
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(PARI) T(n, k)=if(k<0|k>n, 0, prod(i=0, k-1, (2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!))
(PARI) {C(n)=if(n<0, 0, (2*n)!/n!/(n+1)!)}; T(n, k)=if(k<0|k>n, 0, matdet(matrix(k, k, i, j, C(i+j-1+n-k))))
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CROSSREFS
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Columns are A000012, A000108, A005700, A006149, A006150, A006151. Diagonals are A000027, A000330, A006858.
Sequence in context: A097615 A062993 A105556 this_sequence A117396 A125860 A125800
Adjacent sequences: A078917 A078918 A078919 this_sequence A078921 A078922 A078923
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Michael Somos, Dec 15 2002
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