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

1,5

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

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

Sequence in context: A340427 A176627 A015129 * A289673 A010211 A335503

Adjacent sequences: A172373 A172374 A172375 * A172377 A172378 A172379

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved