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

0,1

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A099237.

Sequence in context: A205552 A255707 A260757 * A156786 A156141 A174555

Adjacent sequences: A157111 A157112 A157113 * A157115 A157116 A157117

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 23 2009

Extensions

Edited by G. C. Greubel, Mar 09 2021

Status

approved