A182073 Square array read by antidiagonals: T(n,k) = n!*k! / (floor(n/2)! * floor(k/2)! * floor((n+k)/2)!).

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 6, 3, 2, 3, 6, 30, 6, 6, 6, 6, 30, 20, 10, 4, 6, 4, 10, 20, 140, 20, 20, 12, 12, 20, 20, 140, 70, 35, 10, 15, 6, 15, 10, 35, 70, 630, 70, 70, 30, 30, 30, 30, 70, 70, 630, 252, 126, 28, 42, 12, 30, 12, 42, 28, 126, 252

Offset

0,4

Comments

Compare with A068555 whose entries are given by (2*n)!*(2*k)!/(n!*k!*(n+k)!). See also A211226.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

That T(n,k) is an integer follows from the formulas:

T(2*n,2*k) = (2*n)!*(2*k)!/(n!*k!*(n+k)!) = A068555(n,k);

T(2*n,2*k+1) = (2*n)!*(2*k+1)!/(n!*k!*(n+k)!) = (2*k+1)*A068555(n,k);

T(2*n+1,2*k) = (2*n+1)!*(2*k)!/(n!*k!*(n+k)!) = (2*n+1)*A068555(n,k);

T(2*n+1,2*k+1) = (2*n+1)!*(2*k+1)!/(n!*k!*(n+k+1)!) = (2*k+1)*A135573(n,k).

Example

As a square array
.n\k.|...0....1....2....3....4....5....6...
= = = = = = = = = = = = = = = = = = = = = =
..0..|...1....1....2....6....6...30...20...
..1..|...1....1....2....3....6...10...20...
..2..|...2....2....2....6....4...20...10...
..3..|...6....3....6....6...12...15...30...
..4..|...6....6....4...12....6...30...12...
..5..|..30...10...20...15...30...30...60...
..6..|..20...20...10...30...12...60...20...
...
Formatted as a triangle
.n\k.|...0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = = =
..0..|...1
..1..|...1....1
..2..|...2....1....2
..3..|...6....2....2....6
..4..|...6....3....2....3....6
..5..|..30....6....6....6....6...30
..6..|..20...10....4....6....4...10...20
...

Mathematica

T[n_, k_] := n!*k!/(Floor[n/2]!*Floor[k/2]!*Floor[(n + k)/2]!);
TableForm[Table[T[n, k], {n, 0, 5}, {k, 0, 10}]]
Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* G. C. Greubel, Aug 20 2017 *)

Crossrefs

Cf. A068555, A135573, A211226.

Sequence in context: A258615 A152431 A143965 * A328001 A347563 A349269

Adjacent sequences: A182070 A182071 A182072 * A182074 A182075 A182076

Keyword

nonn,easy,tabl

Author

Peter Bala, Apr 10 2012

Status

approved