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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.

%I A340262 #11 May 18 2024 14:52:54
%S A340262 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,
%T A340262 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,
%U A340262 66,0,286,0,1001,0,3003,1,0,12,0,78,0,364,0,1365,0,4368,0
%N 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.
%e A340262 Triangle starts:
%e A340262                              [0] 1;
%e A340262                            [1] 1, 0;
%e A340262                           [2] 1, 0, 3;
%e A340262                         [3] 1, 0, 4, 0;
%e A340262                       [4] 1, 0, 5, 0, 15;
%e A340262                      [5] 1, 0, 6, 0, 21, 0;
%e A340262                    [6] 1, 0, 7, 0, 28, 0, 84;
%e A340262                  [7] 1, 0, 8, 0, 36, 0, 120, 0;
%e A340262               [8] 1, 0, 9, 0, 45, 0, 165, 0, 495;
%e A340262             [9] 1, 0, 10, 0, 55, 0, 220, 0, 715, 0;
%p A340262 T := proc(n, k) `if`(k::even, combinat:-multinomial(n + k/2, n, k/2), 0) end:
%p A340262 seq(seq(T(n,k), k=0..n), n=0..11);
%t A340262 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A340262 T[n_, k_] := If[EvenQ[k], multinomial[n + k/2, {n, k/2}], 0];
%t A340262 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 18 2024 *)
%Y A340262 Cf. A130746, A067147.
%K A340262 nonn,tabl
%O A340262 0,6
%A A340262 _Peter Luschny_, Jan 05 2021