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

%I A385733 #10 Jul 14 2025 07:11:40
%S A385733 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,
%T A385733 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,
%U A385733 1,1,1,7,77,77,7,1,1,1,1,1,1,3,2,1,11,99,11,1,2,3,1,1
%N A385733 Triangle read by rows: the denominators of the Lucas triangle.
%H A385733 Diana L. Wells, <a href="https://www.fq.math.ca/Scanned/32-2/wells.pdf">The Fibonacci and Lucas triangles modulo 2</a>, Fibonacci Quart. 32, no. 2 (1994), p. 112.
%F A385733 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.
%F A385733 T(n, k) = denominator(LT(n, k)).
%e A385733 Triangle begins:
%e A385733   [0] 1;
%e A385733   [1] 1, 1;
%e A385733   [2] 1, 1, 1;
%e A385733   [3] 1, 1, 1, 1;
%e A385733   [4] 1, 1, 3, 1,  1;
%e A385733   [5] 1, 1, 3, 3,  1, 1;
%e A385733   [6] 1, 1, 1, 2,  1, 1, 1;
%e A385733   [7] 1, 1, 1, 2,  2, 1, 1, 1;
%e A385733   [8] 1, 1, 3, 2, 14, 2, 3, 1, 1;
%e A385733   [9] 1, 1, 3, 3,  7, 7, 3, 3, 1, 1;
%p A385733 c := arccsch(2) - I*Pi/2:
%p A385733 LT := (n, k) -> mul(I^j*cosh(c*j), j = k + 1..n) / mul(I^j*cosh(c*j), j = 1..n - k):
%p A385733 T := (n, k) -> denom(simplify(LT(n, k))): seq(seq(T(n, k), k = 0..n), n = 0..12);
%t A385733 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}]];
%t A385733 Table[Simplify[T[n, k]], {n, 0, 8}, {k, 0, n}] // Flatten // Denominator
%Y A385733 Cf. A385732 (numerators), A070825 (Lucanorial), A003266 (Fibonorial), A010048 (Fibonomial).
%K A385733 nonn,tabl,frac
%O A385733 0,13
%A A385733 _Peter Luschny_, Jul 08 2025