A340262 T(n, k) = multinomial(n + k/2; n, k/2) if k is even else 0. Triangle read by rows, for 0 <= k <= n.

1, 1, 0, 1, 0, 3, 1, 0, 4, 0, 1, 0, 5, 0, 15, 1, 0, 6, 0, 21, 0, 1, 0, 7, 0, 28, 0, 84, 1, 0, 8, 0, 36, 0, 120, 0, 1, 0, 9, 0, 45, 0, 165, 0, 495, 1, 0, 10, 0, 55, 0, 220, 0, 715, 0, 1, 0, 11, 0, 66, 0, 286, 0, 1001, 0, 3003, 1, 0, 12, 0, 78, 0, 364, 0, 1365, 0, 4368, 0

Offset: 0

Author

Peter Luschny, Jan 05 2021

Keywords

• nonn
• tabl

Examples

			Triangle starts:
                             [0] 1;
                           [1] 1, 0;
                          [2] 1, 0, 3;
                        [3] 1, 0, 4, 0;
                      [4] 1, 0, 5, 0, 15;
                     [5] 1, 0, 6, 0, 21, 0;
                   [6] 1, 0, 7, 0, 28, 0, 84;
                 [7] 1, 0, 8, 0, 36, 0, 120, 0;
              [8] 1, 0, 9, 0, 45, 0, 165, 0, 495;
            [9] 1, 0, 10, 0, 55, 0, 220, 0, 715, 0;
		

Crossrefs

Cf. A130746, A067147.

Programs

Maple

T := proc(n, k) `if`(k::even, combinat:-multinomial(n + k/2, n, k/2), 0) end:
seq(seq(T(n,k), k=0..n), n=0..11);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
T[n_, k_] := If[EvenQ[k], multinomial[n + k/2, {n, k/2}], 0];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 18 2024 *)