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URL: https://oeis.org/A189913

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A189913
Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!).
1
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.
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.
Sequence in context: A308028 A356077 A320902 * A240807 A355201 A334347
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Luschny, May 24 2011
STATUS
approved