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