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

0,8

Links

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

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).

Example

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, ...

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 *)

Crossrefs

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

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

Main diagonal gives: A227402.

Cf. A071920.

Sequence in context: A118343 A309148 A351761 * A308460 A244116 A138133

Adjacent sequences: A226028 A226029 A226030 * A226032 A226033 A226034

Keyword

nonn,tabl

Author

Alois P. Heinz, May 23 2013

Status

approved