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A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3*binomial(n+1,k)* binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, read by rows.

%I A281293 #20 Jan 27 2017 13:35:51
%S A281293 1,0,3,0,3,6,0,3,15,10,0,3,27,45,15,0,3,42,126,105,21,0,3,60,280,420,
%T A281293 210,28,0,3,81,540,1260,1134,378,36,0,3,105,945,3150,4410,2646,630,45,
%U A281293 0,3,132,1540,6930,13860,12936,5544,990,55,0,3,162,2376,13860,37422,49896,33264,10692,1485,66
%N A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3*binomial(n+1,k)* binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, read by rows.
%C A281293 The current array is the case m = 2 of the generalized Narayana numbers N_m(n,k) := (m+1)/(n+1)*binomial(n+1,k)*binomial(n-m-1,k-1) for m >= 0, n >= m and 0 <= k <= n-m with N_m(n,0) = A000007(n-m). Case m = 0 gives the table of Narayana numbers A001263 without leading column N_0(n,0) = A000007(n). For m = 1 see A281260, and for m = 3 see A281297.
%H A281293 David Callan, <a href="/A281260/a281260.pdf">Generalized Narayana Numbers</a>
%F A281293 Row sums are A033184(n+1,3).
%e A281293 The triangle begins:
%e A281293 n\k:  0  1    2     3      4      5       6       7      8      9    10  11  ...
%e A281293 02 :  1
%e A281293 03 :  0  3
%e A281293 04 :  0  3    6
%e A281293 05 :  0  3   15    10
%e A281293 06 :  0  3   27    45     15
%e A281293 07 :  0  3   42   126    105     21
%e A281293 08 :  0  3   60   280    420    210      28
%e A281293 09 :  0  3   81   540   1260   1134     378      36
%e A281293 10 :  0  3  105   945   3150   4410    2646     630     45
%e A281293 11 :  0  3  132  1540   6930  13860   12936    5544    990     55
%e A281293 12 :  0  3  162  2376  13860  37422   49896   33264  10692   1485    66
%e A281293 13 :  0  3  195  3510  25740  90090  162162  154440  77220  19305  2145  78
%e A281293 etc.
%t A281293 Table[3 Binomial[n + 1, k] Binomial[n - 3, k - 1]/(n + 1), {n, 2, 12}, {k, 0, n - 2}] // Flatten (* _Michael De Vlieger_, Jan 19 2017 *)
%Y A281293 Cf. A000007, A001263, A033184, A281260, A281297.
%K A281293 nonn,tabl,easy
%O A281293 2,3
%A A281293 _Werner Schulte_, Jan 19 2017