A034801 Triangle of Fibonomial coefficients (k=2).

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

Author

N. J. A. Sloane

Keywords

• nonn
• tabl

Examples

			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;
		

References

• A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 88.

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• 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.

Crossrefs

Cf. A010048.

Programs

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

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 *)

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

Sage

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

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).

Extensions

More terms from James Sellers, Feb 09 2000