OFFSET
0,12
COMMENTS
Start from the sequence 0, 1, 1, 1, 6, 6, 11, 36, 41, 91, 221, 296, 676, 1401, 2156, ..., f(n) = f(n-2) + 5*f(n-3), and its partial products c(n) = 1, 1, 1, 1, 6, 36, 396, 14256, 584496, 53189136, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).
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) = f(n-2, q) + q*f(n-3, q), f(0,q)=0, f(1,q) = f(2,q) = 1, and q = 5. - G. C. Greubel, May 09 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 6, 6, 6, 1;
1, 6, 36, 36, 6, 1;
1, 11, 66, 396, 66, 11, 1;
1, 36, 396, 2376, 2376, 396, 36, 1;
1, 41, 1476, 16236, 16236, 16236, 1476, 41, 1;
1, 91, 3731, 134316, 246246, 246246, 134316, 3731, 91, 1;
1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1;
MATHEMATICA
f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, 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, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 09 2021 *)
PROG
(Sage)
@CachedFunction
def f(n, q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, 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, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Definition corrected to give integral terms by G. C. Greubel, May 09 2021
STATUS
approved