A172428 Triangle T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 3, read by rows.

1, 1, 1, 1, -8, 1, 1, 6, 6, 1, 1, -80, 60, -80, 1, 1, 120, 1200, 1200, 120, 1, 1, -728, 10920, -145600, 10920, -728, 1, 1, 5040, 458640, 9172800, 9172800, 458640, 5040, 1, 1, -6560, 4132800, -501446400, 752169600, -501446400, 4132800, -6560, 1, 1, 362880, 297561600, 249951744000, 2274560870400, 2274560870400, 249951744000, 297561600, 362880, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Formula

T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 3.

Example

Triangle begins as:
  1;
  1,     1;
  1,    -8,       1;
  1,     6,       6,          1;
  1,   -80,      60,        -80,         1;
  1,   120,    1200,       1200,       120,          1;
  1,  -728,   10920,    -145600,     10920,       -728,       1;
  1,  5040,  458640,    9172800,   9172800,     458640,    5040,     1;
  1, -6560, 4132800, -501446400, 752169600, -501446400, 4132800, -6560, 1;

Mathematica

f[n_, q_]:= ((1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n))/2;
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, q): return ((1-q^n)*(1+(-1)^n) + factorial(n)*(1-(-1)^n))/2
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 07 2021

Crossrefs

Cf. A172427 (q=3), this sequence (q=4), A172429 (q=5).

Sequence in context: A359540 A010151 A021556 * A248581 A178163 A197110

Adjacent sequences: A172425 A172426 A172427 * A172429 A172430 A172431

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 02 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved