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A172373
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Triangle T(n, k) = A010048(n, k)*A010048(n, k-1)/Fibonacci(n), read by rows.
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1
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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
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OFFSET
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1,5
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COMMENTS
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These numbers are called the FiboNarayana numbers by Garrett and Killpatrick. - Michel Marcus, Oct 23 2019
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LINKS
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FORMULA
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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