OFFSET
0,5
LINKS
Donald E. Knuth and Herbert S. Wilf, The Power of a Prime that Divides a Generalized Binomial Coefficient, J. Reine Angew. Math. 396 (1989), 212-219.
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics 1 (1878), 184-240, §9.
Diana L. Wells, The Fibonacci and Lucas triangles modulo 2, Fibonacci Quart. 32, no. 2 (1994), p. 112.
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. If in this formula cosh is substituted by sinh one gets the Fibonomial triangle A010048.
T(n, k) = numerator(LT(n, k)).
EXAMPLE
Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 3, 1;
[3] 1, 4, 4, 1;
[4] 1, 7, 28, 7, 1;
[5] 1, 11, 77, 77, 11, 1;
[6] 1, 18, 66, 231, 66, 18, 1;
[7] 1, 29, 174, 957, 957, 174, 29, 1;
[8] 1, 47, 1363, 4089, 44979, 4089, 1363, 47, 1;
[9] 1, 76, 3572, 25897, 155382, 155382, 25897, 3572, 76, 1;
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) -> numer(simplify(LT(n, k))): seq(seq(T(n, k), k = 0..n), n = 0..10);
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 // Numerator
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Jul 08 2025
STATUS
approved