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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.
2
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
3,3
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.
FORMULA
Row sums are A033184(n+1,4).
G.f.: A(x) = x*A281260(x,y)^2. - Vladimir Kruchinin, Oct 10 2020
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Werner Schulte, Jan 19 2017
STATUS
approved