A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.
1, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 2, 16, 38, 1, 1, 2, 24, 146, 192, 1, 1, 2, 32, 326, 1408, 1002, 1, 1, 2, 40, 578, 4672, 14002, 5336, 1, 1, 2, 48, 902, 11008, 69002, 142000, 28814, 1, 1, 2, 56, 1298, 21440, 216002, 1038984, 1459810, 157184, 1, 1, 2, 64, 1766, 36992, 525002, 4320608, 15856206, 15158272, 864146, 1
Offset: 0
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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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 *)
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.
T(n,k) = binomial(k*n-1, n-1)*hypergeom([-n, k*n], [1+(k-1)*n], -1) for k > 0. - Stefano Spezia, Aug 09 2025