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A078920
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Upper triangle of Catalan Number Wall.
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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; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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
| 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
M. Somos, Number Walls in Combinatorics.
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FORMULA
| T(n, k) = Prod[i=1..n-k, Prod[j=i..n-k, (i+j+2n)/(i+j) ]].
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EXAMPLE
| 1; 1,1; 1,2,1; 1,5,3,1; 1,14,14,4,1; ...
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PROG
| (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
| Columns are A000012, A000108, A005700, A006149, A006150, A006151. Diagonals are A000027, A000330, A006858.
Sequence in context: A097615 A062993 A105556 * A186020 A117396 A125860
Adjacent sequences: A078917 A078918 A078919 * A078921 A078922 A078923
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Michael Somos, Dec 15 2002
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