OFFSET
0,3
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)
EXAMPLE
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.
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
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Oct 30 2020
STATUS
approved