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
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 02 2010
EXTENSIONS
Edited by G. C. Greubel, May 07 2021
STATUS
approved