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A281297 Triangular array of generalized Narayana numbers T(n,k) = 4*binomial(n+1,k)* binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.

%I A281297 #19 Oct 14 2020 11:06:09
%S A281297 1,0,4,0,4,10,0,4,24,20,0,4,42,84,35,0,4,64,224,224,56,0,4,90,480,840,
%T A281297 504,84,0,4,120,900,2400,2520,1008,120,0,4,154,1540,5775,9240,6468,
%U A281297 1848,165,0,4,192,2464,12320,27720,29568,14784,3168,220,0,4,234,3744,24024,72072,108108,82368,30888,5148
%N A281297 Triangular array of generalized Narayana numbers T(n,k) = 4*binomial(n+1,k)* binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.
%C A281297 The current array is the case m = 3 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 = 2 see A281293.
%H A281297 David Callan, <a href="/A281260/a281260.pdf">Generalized Narayana Numbers </a>.
%F A281297 Row sums are A033184(n+1,4).
%F A281297 G.f.: A(x) = x*A281260(x,y)^2. - _Vladimir Kruchinin_, Oct 10 2020
%e A281297 The triangle begins:
%e A281297 n\k:  0  1    2     3      4      5       6      7      8     9   10  . . .
%e A281297 03 :  1
%e A281297 04 :  0  4
%e A281297 05 :  0  4   10
%e A281297 06 :  0  4   24    20
%e A281297 07 :  0  4   42    84     35
%e A281297 08 :  0  4   64   224    224     56
%e A281297 09 :  0  4   90   480    840    504      84
%e A281297 10 :  0  4  120   900   2400   2520    1008    120
%e A281297 11 :  0  4  154  1540   5775   9240    6468   1848    165
%e A281297 12 :  0  4  192  2464  12320  27720   29568  14784   3168   220
%e A281297 13 :  0  4  234  3744  24024  72072  108108  82368  30888  5148  286
%e A281297 etc.
%Y A281297 Cf. A000007, A001263, A033184, A281260, A281293.
%K A281297 nonn,tabl,easy
%O A281297 3,3
%A A281297 _Werner Schulte_, Jan 19 2017