OFFSET
0,17
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*f(n-1, q) + f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 1. - G. C. Greubel, May 08 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
1, 2, 2, 2, 2, 1;
1, 3, 6, 6, 6, 3, 1;
1, 4, 12, 24, 24, 12, 4, 1;
1, 5, 20, 60, 120, 60, 20, 5, 1;
1, 7, 35, 140, 420, 420, 140, 35, 7, 1;
1, 10, 70, 350, 1400, 2100, 1400, 350, 70, 10, 1;
MATHEMATICA
f[n_, q_]:= f[n, q]= If[n==0, 0, If[n<4, 1, q*f[n-1, q] + f[n-4, q]]];
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)
PROG
(Sage)
@CachedFunction
def f(n, q): return 0 if (n==0) else 1 if (n<4) else q*f(n-1, q) + f(n-4, q)
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Definition corrected to give integral terms, G. C. Greubel, May 08 2021
STATUS
approved