A385733 Triangle read by rows: the denominators of the Lucas triangle.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 14, 2, 3, 1, 1, 1, 1, 3, 3, 7, 7, 3, 3, 1, 1, 1, 1, 1, 1, 7, 77, 7, 1, 1, 1, 1, 1, 1, 1, 1, 7, 77, 77, 7, 1, 1, 1, 1, 1, 1, 3, 2, 1, 11, 99, 11, 1, 2, 3, 1, 1

Offset: 0

Author

Peter Luschny, Jul 08 2025

Keywords

• nonn
• tabl
• frac

Examples

			Triangle begins:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 1, 1;
  [4] 1, 1, 3, 1,  1;
  [5] 1, 1, 3, 3,  1, 1;
  [6] 1, 1, 1, 2,  1, 1, 1;
  [7] 1, 1, 1, 2,  2, 1, 1, 1;
  [8] 1, 1, 3, 2, 14, 2, 3, 1, 1;
  [9] 1, 1, 3, 3,  7, 7, 3, 3, 1, 1;
		

Links

• Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), p. 112.

Crossrefs

Cf. A385732 (numerators), A070825 (Lucanorial), A003266 (Fibonorial), A010048 (Fibonomial).

Programs

Maple

c := arccsch(2) - I*Pi/2:
LT := (n, k) -> mul(I^j*cosh(c*j), j = k + 1..n) / mul(I^j*cosh(c*j), j = 1..n - k):
T := (n, k) -> denom(simplify(LT(n, k))): seq(seq(T(n, k), k = 0..n), n = 0..12);

Mathematica

T[n_, k_] := With[{c = ArcCsch[2] - I Pi/2}, Product[I^j Cosh[c j], {j, k + 1, n}] / Product[I^j Cosh[c j], {j, 1, n - k}]];
Table[Simplify[T[n, k]], {n, 0, 8}, {k, 0, n}] // Flatten // Denominator

Formula

LT(n, k) = Product_{j=k+1..n} i^j*cosh(c*j) / Product_{j=1..n-k} i^j*cosh(c*j) where c = arccsch(2) - i*Pi/2 and i is the imaginary unit.

T(n, k) = denominator(LT(n, k)).