OFFSET
1,5
COMMENTS
These numbers are called the FiboNarayana numbers by Garrett and Killpatrick. - Michel Marcus, Oct 23 2019
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Kristina Garrett and Kendra Killpatrick, A Recursion for the FiboNarayana and the Generalized Narayana Numbers, arXiv:1910.08855 [math.CO], 2019.
FORMULA
T(n, k) = A010048(n, k) * A010048(n, k-1) / Fibonacci(n) where A010048 are the Fibonomial coefficients. - Michel Marcus, Oct 23 2019
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 6, 6, 1;
1, 15, 45, 15, 1;
1, 40, 300, 300, 40, 1;
1, 104, 2080, 5200, 2080, 104, 1;
1, 273, 14196, 94640, 94640, 14196, 273, 1;
1, 714, 97461, 1689324, 4504864, 1689324, 97461, 714, 1;
1, 1870, 667590, 30375345, 210602392, 210602392, 30375345, 667590, 1870, 1;
MATHEMATICA
A010048[n_, k_]:= Product[Fibonacci[n-j+1]/Fibonacci[j], {j, k}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, May 08 2021 *)
PROG
(PARI) fibonomial(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)); \\ A010048
T(n, k) = fibonomial(n, k)*fibonomial(n, k-1)/fibonacci(n);
matrix(10, 10, n, k, T(n, k)) \\ to see the triangle \\ Michel Marcus, Oct 23 2019
(Sage)
@CachedFunction
def A010048(n, q): return product( fibonacci(n-j+1)/fibonacci(j) for j in (1..k) )
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 08 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Edited by Michel Marcus and Joerg Arndt, Oct 23 2019
STATUS
approved