A189913 Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!).

1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 1, 4, 6, 12, 2, 1, 5, 10, 30, 10, 10, 1, 6, 15, 60, 30, 60, 5, 1, 7, 21, 105, 70, 210, 35, 35, 1, 8, 28, 168, 140, 560, 140, 280, 14, 1, 9, 36, 252, 252, 1260, 420, 1260, 126, 126, 1, 10, 45, 360, 420, 2520, 1050, 4200, 630, 1260, 42

Offset: 0

Author

Peter Luschny, May 24 2011

Keywords

• nonn
• tabl
• easy

Comments

The triangle may be regarded a generalization of the triangle A097610:

A097610(n,k) = binomial(n,k)*(2*k)$/(k+1);

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

Here n$ denotes the swinging factorial A056040(n). As A097610 is a decomposition of the Motzkin numbers A001006, 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, 2,  1
[3]  1, 3,  3,   3
[4]  1, 4,  6,  12,  2
[5]  1, 5, 10,  30, 10,  10
[6]  1, 6, 15,  60, 30,  60,  5
[7]  1, 7, 21, 105, 70, 210, 35, 35
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
• Peter Luschny, The lost Catalan numbers.

Crossrefs

Row sums are A189912.

Cf. A097610, A057977, A001006.

Programs

Magma

/* As triangle */ [[Binomial(n,k)*Factorial(k)/(Factorial(Floor(k/2))*Factorial(Floor((k + 2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 13 2018

Maple

A189913 := (n,k) -> binomial(n,k)*(k!/iquo(k,2)!^2)/(iquo(k,2)+1):
seq(print(seq(A189913(n,k),k=0..n)),n=0..7);

Mathematica

T[n_, k_] := Binomial[n, k]*k!/((Floor[k/2])!*(Floor[(k + 2)/2])!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* G. C. Greubel, Jan 13 2018 *)

PARI

{T(n,k) = binomial(n,k)*k!/((floor(k/2))!*(floor((k+2)/2))!) };
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 13 2018

Formula

From R. J. Mathar, Jun 07 2011: (Start)

T(n,1) = n.

T(n,2) = A000217(n-1).

T(n,3) = A027480(n-2).

T(n,4) = A034827(n). (End)