OFFSET
0,8
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=0..n} v(j, q)((1)), v(n, q) = M*v(n-1, q), v(0, q) = {1, 1, 1}, M = {{0, 1, 0}, {0, 0, 1}, {q^3, q^3, 0}}, and q = 2.
T(n, k, q) = round( (1/f(k,q))*Product_{j=0..n-k} f(j+k,q)/f(j,q) ), where f(n, q) = 6*q*f(n-2, q) + 8*q^3*f(n-3, q), and f(0,q) = f(1,q) = f(2,q) = 1, and q = 2. - G. C. Greubel, Jul 06 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 16, 16, 1;
1, 16, 256, 16, 1;
1, 136, 2176, 2176, 136, 1;
1, 256, 34816, 34816, 34816, 256, 1;
1, 1216, 311296, 2646016, 2646016, 311296, 1216, 1;
1, 3136, 3813376, 61014016, 518619136, 61014016, 3813376, 3136, 1;
MATHEMATICA
f[n_, q_]:= f[n, q] = If[n<3, 1, q^3*f[n-2, q] + q^3*f[n-3, q]];
c[n_, q_]:= Product[f[j, q], {j, 0, n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)
PROG
(Magma)
function f(n, k)
if n lt 3 then return 1;
else return k^3*f(n-2, k) + k^3*f(n-3, k);
end if; return f;
end function;
T:= func< n, k, q | Round( (&*[f(j+k, q)/f(j, q): j in [0..n-k]])/f(k, q) ) >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021
(Sage)
@CachedFunction
def f(n, q): return 1 if (n<3) else q^3*f(n-2, q) + q^3*f(n-3, q)
def T(n, k, q): return round( product( f(j+k, q)/f(j, q) for j in (0..n-k))/f(k, q) )
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 24 2010
EXTENSIONS
Definition corrected and edited by G. C. Greubel, Jul 06 2021
STATUS
approved