OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..30 of the triangle, flattened
FORMULA
T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q)), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = q*(f(n-1, q) + f(n-2, q)), f(0, q) = 0, f(1, q) = 1, and q = 3.
T(n, k, q) = c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k,q)), where c(n, q) = Product_{j=1..n} f(j, q), and f(n, q) = (-I*sqrt(q))^(n-1)*ChebyshevU(n-1, i*sqrt(q)/2). - G. C. Greubel, May 07 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 12, 1;
1, 180, 180, 1;
1, 2565, 38475, 2565, 1;
1, 36936, 7895070, 7895070, 36936, 1;
1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1;
MATHEMATICA
f[n_, q_]:= (-I*Sqrt[q])^(n-1)*ChebyshevU[n-1, I*Sqrt[q]/2];
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= c[n-1, q]*c[n, q]/(c[k-1, q]^2*c[n-k, q]*c[n-k+1, q]*f[k, q]);
Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)
PROG
(Sage)
@CachedFunction
def f(n, q): return (-i*sqrt(q))^(n-1)*chebyshev_U(n-1, i*sqrt(q)/2)
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return c(n-1, q)*c(n, q)/(c(k-1, q)^2*c(n-k, q)*c(n-k+1, q)*f(k, q))
flatten([[T(n, k, 3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 07 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Edited by G. C. Greubel, May 07 2021
STATUS
approved