A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 16, 22, 0, 1, 4, 36, 161, 130, 0, 1, 5, 64, 525, 1716, 791, 0, 1, 6, 100, 1222, 8086, 18832, 4900, 0, 1, 7, 144, 2360, 24616, 128248, 210574, 30738, 0, 1, 8, 196, 4047, 58730, 510664, 2072862, 2385644, 194634, 0

Offset: 0

Author

Alois P. Heinz, May 23 2013

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  1,   1,     1,      1,      1,       1, ...
  0,   1,     2,      3,      4,       5, ...
  0,   4,    16,     36,     64,     100, ...
  0,  22,   161,    525,   1222,    2360, ...
  0, 130,  1716,   8086,  24616,   58730, ...
  0, 791, 18832, 128248, 510664, 1505205, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-2 give: A000007, A088536, A226012.

Rows n=0-2 give: A000012, A001477, A016742.

Main diagonal gives: A227402.

Cf. A071920.

Programs

Maple

A:= (n, k)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..k*n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

A[n_, k_] := If[n==0, 1, Sum[Binomial[n + 2j - 1, 2j], {j, 0, k n - 1}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{j=0..k*n-1} C(n+2*j-1,2*j), A(0,k) = 1.

A(n,k) = A071921(n,k*n).