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.

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, 504, 84, 0, 4, 120, 900, 2400, 2520, 1008, 120, 0, 4, 154, 1540, 5775, 9240, 6468, 1848, 165, 0, 4, 192, 2464, 12320, 27720, 29568, 14784, 3168, 220, 0, 4, 234, 3744, 24024, 72072, 108108, 82368, 30888, 5148

Offset: 3

Author

Werner Schulte, Jan 19 2017

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			The triangle begins:
n\k:  0  1    2     3      4      5       6      7      8     9   10  . . .
03 :  1
04 :  0  4
05 :  0  4   10
06 :  0  4   24    20
07 :  0  4   42    84     35
08 :  0  4   64   224    224     56
09 :  0  4   90   480    840    504      84
10 :  0  4  120   900   2400   2520    1008    120
11 :  0  4  154  1540   5775   9240    6468   1848    165
12 :  0  4  192  2464  12320  27720   29568  14784   3168   220
13 :  0  4  234  3744  24024  72072  108108  82368  30888  5148  286
etc.
		

Links

• David Callan, Generalized Narayana Numbers .

Crossrefs

Cf. A000007, A001263, A033184, A281260, A281293.

Formula

Row sums are A033184(n+1,4).

G.f.: A(x) = x*A281260(x,y)^2. - Vladimir Kruchinin, Oct 10 2020