A172376 Triangle 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, read by rows.

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, 1, 7628985, 337399490610, 69234375473172, 69234375473172, 337399490610, 7628985, 1, 1, 109656180, 69713779364775, 205544107079102610, 2959835141939077584, 205544107079102610, 69713779364775, 109656180, 1

Offset: 1

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Crossrefs

Cf. A002605, A030195, A172375 (q=2), this sequence (q=3).

Programs

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 *)

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

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

Extensions

Edited by G. C. Greubel, May 07 2021