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%I #15 May 09 2021 06:10:40
%S 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,
%T 2080,104,1,1,273,14196,94640,94640,14196,273,1,1,714,97461,1689324,
%U 4504864,1689324,97461,714,1,1,1870,667590,30375345,210602392,210602392,30375345,667590,1870,1
%N Triangle T(n, k) = A010048(n, k)*A010048(n, k-1)/Fibonacci(n), read by rows.
%C These numbers are called the FiboNarayana numbers by Garrett and Killpatrick. - _Michel Marcus_, Oct 23 2019
%H G. C. Greubel, <a href="/A172373/b172373.txt">Rows n = 1..50 of the triangle, flattened</a>
%H Kristina Garrett and Kendra Killpatrick, <a href="https://arxiv.org/abs/1910.08855">A Recursion for the FiboNarayana and the Generalized Narayana Numbers</a>, arXiv:1910.08855 [math.CO], 2019.
%F T(n, k) = A010048(n, k) * A010048(n, k-1) / Fibonacci(n) where A010048 are the Fibonomial coefficients. - _Michel Marcus_, Oct 23 2019
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 6, 6, 1;
%e 1, 15, 45, 15, 1;
%e 1, 40, 300, 300, 40, 1;
%e 1, 104, 2080, 5200, 2080, 104, 1;
%e 1, 273, 14196, 94640, 94640, 14196, 273, 1;
%e 1, 714, 97461, 1689324, 4504864, 1689324, 97461, 714, 1;
%e 1, 1870, 667590, 30375345, 210602392, 210602392, 30375345, 667590, 1870, 1;
%t A010048[n_, k_]:= Product[Fibonacci[n-j+1]/Fibonacci[j], {j, k}];
%t T[n_, k_]:= A010048[n, k]*A010048[n, k-1]/Fibonacci[n];
%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, May 08 2021 *)
%o (PARI) fibonomial(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)); \\ A010048
%o T(n, k) = fibonomial(n,k)*fibonomial(n, k-1)/fibonacci(n);
%o matrix(10, 10, n, k, T(n, k)) \\ to see the triangle \\ _Michel Marcus_, Oct 23 2019
%o (Sage)
%o @CachedFunction
%o def A010048(n,q): return product( fibonacci(n-j+1)/fibonacci(j) for j in (1..k) )
%o def T(n,k,q): return A010048(n,k)*A010048(n,k-1)/fibonacci(n)
%o flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, May 08 2021
%Y Cf. A010048 (Fibonomial coefficients), A001263 (Narayana numbers).
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_, Feb 01 2010
%E Edited by _Michel Marcus_ and _Joerg Arndt_, Oct 23 2019
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