A190909 - OEIS

A190909

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

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 6, 1, 15, 70, 168, 54, 30, 1, 21, 140, 504, 270, 330, 20, 1, 28, 252, 1260, 990, 1980, 260, 140, 1, 36, 420, 2772, 2970, 8580, 1820, 2100, 70, 1, 45, 660, 5544, 7722, 30030, 9100, 16800, 1190, 630

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OFFSET

0,5

COMMENTS

The triangle may be regarded as a generalization of the triangle A063007.

A063007(n,k) = binomial(n+k, n-k)*(2*k)$;

T(n,k) = binomial(n+k, n-k)*(k)$.

Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.

T(n,n) = A056040(n) which can be seen as extended central binomial numbers.

LINKS

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

Peter Luschny, The lost Catalan numbers

R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

FORMULA

T(n,1) = A000217(n). T(n,2) = 2*binomial(n+2,4) (Cf. A034827).

EXAMPLE

[0] 1

[1] 1, 1

[2] 1, 3, 2

[3] 1, 6, 10, 6