A176647 Triangle T(n, k) = f(n, k) + f(n, n-k) - f(n, n), where f(n, k) = binomial(n*(3*n-1)/2 + k, k), read by rows.

1, 1, 1, 1, -9, 1, 1, -351, -351, 1, 1, -12627, -14398, -12627, 1, 1, -575721, -648906, -648906, -575721, 1, 1, -32468384, -35945819, -36238644, -35945819, -32468384, 1, 1, -2186189329, -2387546394, -2403595518, -2403595518, -2387546394, -2186189329, 1

Offset

0,5

Links

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

Formula

T(n, k) = f(n, k) + f(n, n-k) - f(n, n), where f(n, k) = binomial(n*(3*n-1)/2 + k, k).

Example

Triangle begins as:
  1;
  1,         1;
  1,        -9,         1;
  1,      -351,      -351,         1;
  1,    -12627,    -14398,    -12627,         1;
  1,   -575721,   -648906,   -648906,   -575721,         1;
  1, -32468384, -35945819, -36238644, -35945819, -32468384, 1;

Mathematica

f[n_, k_]:= Binomial[n*(3*n-1)/2 + k, k];
T[n_, k_]= f[n, k] + f[n, n-k] - f[n, n];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jul 02 2021 *)

Prog

(Magma)
f:= func< n, k | Binomial(Floor(Binomial(3*n, 2)/3) + k, k) >;
[f(n, k) +f(n, n-k) -f(n, n): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 02 2021
(Sage)
def f(n, k): return binomial(n*(3*n-1)/2 +k, k)
flatten([[f(n, k) + f(n, n-k) - f(n, n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 02 2021

Crossrefs

Cf. A000326, A176567.

Sequence in context: A022172 A173005 A015123 * A068452 A021527 A257437

Adjacent sequences: A176644 A176645 A176646 * A176648 A176649 A176650

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 22 2010

Extensions

Edited by G. C. Greubel, Jul 02 2021

Status

approved