A190907 - OEIS

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

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OFFSET

0,5

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.

LINKS

Table of n, a(n) for n=0..54.

Peter Luschny, The lost Catalan numbers.

FORMULA

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

EXAMPLE

[0] 1

[1] 1, 1

[2] 1, 3, 1

[3] 1, 6, 5, 3

[4] 1, 10, 15, 21, 2