A172373 Triangle T(n, k) = A010048(n, k)*A010048(n, k-1)/Fibonacci(n), read by rows.

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

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}];
T[n_, k_]:= A010048[n, k]*A010048[n, k-1]/Fibonacci[n];
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) )
def T(n, k, q): return A010048(n, k)*A010048(n, k-1)/fibonacci(n)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 08 2021

Crossrefs

Cf. A010048 (Fibonomial coefficients), A001263 (Narayana numbers).

Sequence in context: A223257 A173881 A329228 * A174411 A322620 A155795

Adjacent sequences: A172370 A172371 A172372 * A172374 A172375 A172376

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2010

Extensions

Edited by Michel Marcus and Joerg Arndt, Oct 23 2019

Status

approved