A176567 Triangle T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n), read by rows.

1, 1, 1, 1, 1, 1, 1, -6, -6, 1, 1, -119, -154, -119, 1, 1, -1991, -2651, -2651, -1991, 1, 1, -38744, -50252, -52632, -50252, -38744, 1, 1, -888008, -1118007, -1169366, -1169366, -1118007, -888008, 1, 1, -23535791, -28915001, -30018509, -30188420, -30018509, -28915001, -23535791, 1

Offset

0,8

Links

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

Formula

T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n).

Example

Triangle begins as:
  1;
  1,       1;
  1,       1,        1;
  1,      -6,       -6,        1;
  1,    -119,     -154,     -119,        1;
  1,   -1991,    -2651,    -2651,    -1991,        1;
  1,  -38744,   -50252,   -52632,   -50252,   -38744,       1;
  1, -888008, -1118007, -1169366, -1169366, -1118007, -888008, 1;

Mathematica

T[n_, k_]= Binomial[Binomial[n, 2] + k, k] + Binomial[Binomial[n, 2] + n-k, n-k] - Binomial[Binomial[n, 2] + n, n];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
T:= func< n, k | Binomial(Binomial(n, 2) +k, k) + Binomial(Binomial(n, 2) +n-k, n-k) - Binomial(Binomial(n, 2) +n, n) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
(Sage)
def f(n, k): return binomial(binomial(n, 2) + k, k)
def T(n, k): return f(n, k) + f(n, n-k) - f(n, n)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

Crossrefs

Sequence in context: A155868 A322622 A176565 * A372272 A283100 A065493

Adjacent sequences: A176564 A176565 A176566 * A176568 A176569 A176570

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 22 2010

Extensions

Edited by G. C. Greubel, Jul 09 2021

Status

approved