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A078920 Upper triangle of Catalan Number Wall. +0
4
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)
OFFSET

1,4

COMMENT

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

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.

FORMULA

T(n, k) = Prod[i=1..n-k, Prod[j=i..n-k, (i+j+2n)/(i+j) ]].

EXAMPLE

1; 1,1; 1,2,1; 1,5,3,1; 1,14,14,4,1; ...

PROGRAM

(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))))

CROSSREFS

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

KEYWORD

easy,nonn,tabl

AUTHOR

Michael Somos, Dec 15 2002

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Last modified December 21 10:15 EST 2009. Contains 171081 sequences.


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