OFFSET
0,12
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-3,q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 1.
T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3). - G. C. Greubel, May 09 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 3, 6, 6, 3, 1;
1, 4, 12, 24, 12, 4, 1;
1, 6, 24, 72, 72, 24, 6, 1;
1, 9, 54, 216, 324, 216, 54, 9, 1;
1, 13, 117, 702, 1404, 1404, 702, 117, 13, 1;
1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1;
MATHEMATICA
f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], q*f[n-1, 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, 1], {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 q*f(n-1, 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, 1) 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