A190910 -id:A190910 - cp's OEIS frontend

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

Offset: 0

Peter Luschny, May 24 2011

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.

			[0]  1
[1]  1,  1
[2]  1,  3,   2
[3]  1,  6,  10,    6
[4]  1, 10,  30,   42,   6
[5]  1, 15,  70,  168,  54,   30
[6]  1, 21, 140,  504, 270,  330,  20
[7]  1, 28, 252, 1260, 990, 1980, 260, 140

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.

Cf. Row sums: A190910; A056040, A063007, A085478, A088617.

A190909 := (n,k) -> binomial(n+k,n-k)*k!/iquo(k,2)!^2:
seq(print(seq(A190909(n,k),k=0..n)),n=0..7);

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 25 2012 *)

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

cp's OEIS Frontend

A190909 Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

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