A190907 Triangle read by rows: T(n,k) = binomial(n+k, n-k) k! / (floor(k/2)! * floor((k+2)/2)!).

1, 1, 1, 1, 3, 1, 1, 6, 5, 3, 1, 10, 15, 21, 2, 1, 15, 35, 84, 18, 10, 1, 21, 70, 252, 90, 110, 5, 1, 28, 126, 630, 330, 660, 65, 35, 1, 36, 210, 1386, 990, 2860, 455, 525, 14, 1, 45, 330, 2772, 2574, 10010, 2275, 4200, 238, 126

Offset: 0

Author

Peter Luschny, May 24 2011

Keywords

• nonn
• tabl

Comments

The triangle may be regarded as a generalization of the triangle A088617.

A088617(n,k) = binomial(n+k,n-k)*(2*k)$/(k+1);

T(n,k) = binomial(n+k,n-k)*(k)$ /(floor(k/2)+1).

Here n$ denotes the swinging factorial A056040(n). As A088617 is a decomposition of the large Schroeder numbers A006318, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.

T(n,n) = A057977(n) which can be seen as extended Catalan numbers.

Examples

			[0]  1
[1]  1,  1
[2]  1,  3,   1
[3]  1,  6,   5,   3
[4]  1, 10,  15,  21,   2
[5]  1, 15,  35,  84,  18,  10
[6]  1, 21,  70, 252,  90, 110,  5
[7]  1, 28, 126, 630, 330, 660, 65, 35
		

Links

• Peter Luschny, The lost Catalan numbers.

Crossrefs

Cf. Row sums: A190908; A056040, A085478, A088617, A060693.

Programs

Maple

A190907 := (n,k) -> binomial(n+k,n-k)*k!/(floor(k/2)!*floor((k+2)/2)!);
seq(print(seq(A190907(n,k), k=0..n)), n=0..7);

Mathematica

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!Floor[(k+2)/2]!),{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 05 2012 *)

Formula

T(n,1) = A000217(n). T(n,2) = (n-1)*n*(n+1)*(n+2)/24 (Cf. A000332).