A337841 Triangle read by rows, T(n, k) = binomial(2*n-1, 2*k-1) * binomial(2*n-2*k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.

1, 0, 2, 0, 3, 3, 0, 6, 10, 4, 0, 14, 30, 21, 5, 0, 36, 90, 84, 36, 6, 0, 99, 275, 308, 180, 55, 7, 0, 286, 858, 1092, 780, 330, 78, 8, 0, 858, 2730, 3822, 3150, 1650, 546, 105, 9, 0, 2652, 8840, 13328, 12240, 7480, 3094, 840, 136, 10, 0, 8398, 29070, 46512, 46512, 31977, 15561, 5320, 1224, 171, 11

Offset: 0

Author

Werner Schulte, Oct 30 2020

Keywords

• nonn
• easy
• tabl

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
  n\k:  0     1      2      3      4      5      6     7     8    9  10
=======================================================================
   0 :  1
   1 :  0     2
   2 :  0     3      3
   3 :  0     6     10      4
   4 :  0    14     30     21      5
   5 :  0    36     90     84     36      6
   6 :  0    99    275    308    180     55      7
   7 :  0   286    858   1092    780    330     78     8
   8 :  0   858   2730   3822   3150   1650    546   105     9
   9 :  0  2652   8840  13328  12240   7480   3094   840   136   10
  10 :  0  8398  29070  46512  46512  31977  15561  5320  1224  171  11
etc.
		

Crossrefs

Cf. Row sums: A000984, main diagonal: A000027, 1st subdiagonal: A014105, 2nd subdiagonal: A055112, column 0: A000007, column 1: A007054.

Programs

Maple

T := proc(n, k) option remember; if k = n then n+1 else
T(n-1, k)*(2*n-2)*(2*n-1)/((n-1)*(n+2)-(k-1)*(k+2)) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 02 2020

Formula

T(n,k) = binomial(2*n-1,n-k) * k * (2*k+1) * (2*k+2) / ((n+k)*(n+k+1)) for 1 <= k <= n, and T(n,0) = 0^n for n >= 0.

T(n,n) = n+1 for n >= 0; T(n,n-1) = (n-1) * (2*n-1) for n > 0; T(n,n-2) = (n-1) * (n-2) * (2*n-3) for n > 1.

T(n,k) = T(n-1,k) * (2*n-2) * (2*n-1) / ((n-1) * (n+2) - (k-1) * (k+2)) for 0 <= k < n with initial values T(n,n) = n+1 for n >= 0.

Row sums are A000984(n) for n >= 0.

Alternating row sums are 0 for n > 1.

Sum_{k=0..n} (-1)^k * T(n,k) * (k*(k+1)/2)^m = 0 for 0 <= m <= n-2.

T(n,1) = 12 * binomial(2*n-1,n-1)/((n+1)*(n+2)) = A007054(n) for n > 0.

T(n,k) = T(n,1)*(k*(k+1)*(2*k+1)/6)*binomial(n-1,k-1)/binomial(n+1+k,k-1) for 1 <= k <= n.

From Werner Schulte, Nov 09 2020: (Start)

T(n,k) = A128899(n,k) * (k+1) * (2*k+1) / (n+k+1) for 0 <= k <= n.

T(n,0) + Sum_{k=1..n} T(n,k) / (k*(k+1)) = A000108(n) for n >= 0. (End)