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A034801
Triangle of Fibonomial coefficients (k=2).
8
1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 21, 56, 21, 1, 1, 55, 385, 385, 55, 1, 1, 144, 2640, 6930, 2640, 144, 1, 1, 377, 18096, 124410, 124410, 18096, 377, 1, 1, 987, 124033, 2232594, 5847270, 2232594, 124033, 987, 1, 1, 2584, 850136, 40062659, 274715376, 274715376, 40062659, 850136, 2584, 1
OFFSET
0,5
REFERENCES
A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 88.
LINKS
C. Pita, On s-Fibonomials, J. Int. Seq. 14 (2011) # 11.3.7.
C. J. Pita Ruiz Velasco, Sums of Products of s-Fibonacci Polynomial Sequences, J. Int. Seq. 14 (2011) # 11.7.6.
FORMULA
Fibonomial coefficients formed from sequence F_3k [ 2, 8, 34, ... ].
T(n, k) = Product_{j=0..k-1} Fibonacci(2*(n-j)) / Product_{j=1..k} Fibonacci(2*j).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 8, 8, 1;
1, 21, 56, 21, 1;
1, 55, 385, 385, 55, 1;
1, 144, 2640, 6930, 2640, 144, 1;
1, 377, 18096, 124410, 124410, 18096, 377, 1;
1, 987, 124033, 2232594, 5847270, 2232594, 124033, 987, 1;
MAPLE
A034801 := proc(n, k)
mul(combinat[fibonacci](2*n-2*j), j=0..k-1) /
mul(combinat[fibonacci](2*j), j=1..k) ;
end proc: # R. J. Mathar, Sep 02 2017
MATHEMATICA
F[n_, k_, q_]:= Product[Fibonacci[q*(n-j+1)]/Fibonacci[q*j], {j, k}];
Table[F[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 13 2019 *)
PROG
(PARI) F(n, k, q) = f=fibonacci; prod(j=1, k, f(q*(n-j+1))/f(q*j)); \\ G. C. Greubel, Nov 13 2019
(SageMath)
def F(n, k, q):
if (n==0 and k==0): return 1
else: return product(fibonacci(q*(n-j+1))/fibonacci(q*j) for j in (1..k))
[[F(n, k, 2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 13 2019
(GAP)
F:= function(n, k, q)
if n=0 and k=0 then return 1;
else return Product([1..k], j-> Fibonacci(q*(n-j+1))/Fibonacci(q*j));
fi;
end;
Flat(List([0..10], n-> List([0..n], k-> F(n, k, 2) ))); # G. C. Greubel, Nov 13 2019
CROSSREFS
Cf. A010048.
Sequence in context: A157210 A359573 A359575 * A331890 A102435 A340882
KEYWORD
nonn,tabl
EXTENSIONS
More terms from James Sellers, Feb 09 2000
STATUS
approved