A157114 Triangle T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k), read by rows.

2, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 56, 16, 1, 1, 25, 225, 225, 25, 1, 1, 36, 771, 1632, 771, 36, 1, 1, 49, 2597, 9261, 9261, 2597, 49, 1, 1, 64, 9136, 52384, 71920, 52384, 9136, 64, 1, 1, 81, 33777, 320814, 525987, 525987, 320814, 33777, 81, 1, 1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1

Offset: 0

Author

Roger L. Bagula, Feb 23 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  2;
  1,   1;
  1,   4,      1;
  1,   9,      9,       1;
  1,  16,     56,      16,       1;
  1,  25,    225,     225,      25,       1;
  1,  36,    771,    1632,     771,      36,       1;
  1,  49,   2597,    9261,    9261,    2597,      49,       1;
  1,  64,   9136,   52384,   71920,   52384,    9136,      64,      1;
  1,  81,  33777,  320814,  525987,  525987,  320814,   33777,     81,   1;
  1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A099237.

Programs

Magma

A157114:= func< n,k | Binomial(n*k, n-k) + Binomial(n*(n-k), k) >;
[A157114(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

Maple

A157114:= (n,k) -> binomial(n*k, n-k) + binomial(n*(n-k), k);
seq(seq(A157114(n,k), k=0..n), n=0..12); # G. C. Greubel, Mar 09 2021

Mathematica

T[n_, k_]:= Binomial[n*k, n-k], Binomial[n*(n-k), k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 09 2021 *)

Sage

def A157114(n,k): return binomial(n*k, n-k) + binomial(n*(n-k), k)
flatten([[A157114(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

Formula

T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k).

Sum_{k=0..n} T(n,k) = 2*A099237(n). - G. C. Greubel, Mar 09 2021

Extensions

Edited by G. C. Greubel, Mar 09 2021