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.

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.

Links

Table of n, a(n) for n=2..67.

David Callan, Generalized Narayana Numbers

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

Cf. A000007, A001263, A033184, A281260, A281297.

Sequence in context: A194480 A194485 A120987 * A258108 A282610 A011076

Adjacent sequences: A281290 A281291 A281292 * A281294 A281295 A281296

Keyword

nonn,tabl,easy

Author

Werner Schulte, Jan 19 2017

Status

approved