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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.
2
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, 210, 28, 0, 3, 81, 540, 1260, 1134, 378, 36, 0, 3, 105, 945, 3150, 4410, 2646, 630, 45, 0, 3, 132, 1540, 6930, 13860, 12936, 5544, 990, 55, 0, 3, 162, 2376, 13860, 37422, 49896, 33264, 10692, 1485, 66
OFFSET
2,3
COMMENTS
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.
FORMULA
Row sums are A033184(n+1,3).
EXAMPLE
The triangle begins:
n\k: 0 1 2 3 4 5 6 7 8 9 10 11 ...
02 : 1
03 : 0 3
04 : 0 3 6
05 : 0 3 15 10
06 : 0 3 27 45 15
07 : 0 3 42 126 105 21
08 : 0 3 60 280 420 210 28
09 : 0 3 81 540 1260 1134 378 36
10 : 0 3 105 945 3150 4410 2646 630 45
11 : 0 3 132 1540 6930 13860 12936 5544 990 55
12 : 0 3 162 2376 13860 37422 49896 33264 10692 1485 66
13 : 0 3 195 3510 25740 90090 162162 154440 77220 19305 2145 78
etc.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Werner Schulte, Jan 19 2017
STATUS
approved