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

0,5

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.

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Peter Luschny, The lost Catalan numbers.

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)

Example

[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

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 *)

Prog

(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
(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

Crossrefs

Row sums are A189912.

Cf. A097610, A057977, A001006.

Sequence in context: A308028 A356077 A320902 * A240807 A355201 A334347

Adjacent sequences: A189910 A189911 A189912 * A189914 A189915 A189916

Keyword

nonn,tabl,easy

Author

Peter Luschny, May 24 2011

Status

approved