OFFSET
0,5
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
T(n,k) = (1/k) * [x^n] ( (1 + x)/(1 - x) )^(k*n) for k > 0 and n > 0.
T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j-1,n-1).
T(n,k) = (1/k) * Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j-1,j) for k > 0 and n > 0.
T(n,k) = Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n-1,j-1) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, ...
1, 8, 16, 24, 32, 40, ...
1, 38, 146, 326, 578, 902, ...
1, 192, 1408, 4672, 11008, 21440, ...
1, 1002, 14002, 69002, 216002, 525002, ...
MATHEMATICA
T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n + j - 1, n - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 24 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 24 2020
STATUS
approved