cp's OEIS Frontend

A034802 Triangle of Fibonomial coefficients (k=3).

%I A034802 #20 Jul 02 2025 16:01:56
%S A034802 1,1,1,1,4,1,1,17,17,1,1,72,306,72,1,1,305,5490,5490,305,1,1,1292,
%T A034802 98515,417240,98515,1292,1,1,5473,1767779,31716035,31716035,1767779,
%U A034802 5473,1,1,23184,31721508,2410834608,10212563270,2410834608,31721508,23184,1
%N A034802 Triangle of Fibonomial coefficients (k=3).
%D A034802 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 88.
%H A034802 G. C. Greubel, <a href="/A034802/b034802.txt">Rows n = 0..75 of triangle, flattened</a>
%H A034802 C. Pita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, J. Int. Seq. 14 (2011) # 11.3.7.
%F A034802 T(n, k) = Product_{j=0..k-1} Fibonacci(3*(n-j))/Product_{j=1..k} Fibonacci(3*j).
%F A034802 Fibonomial coefficients formed from sequence F_4k [ 3 21 144 987 ... ].
%t A034802 F[n_, k_, q_]:= Product[Fibonacci[q*(n-j+1)]/Fibonacci[q*j], {j,k}];
%t A034802 Table[F[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 13 2019 *)
%o A034802 (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
%o A034802 (Sage)
%o A034802 def F(n,k,q):
%o A034802     if (n==0 and k==0): return 1
%o A034802     else: return product(fibonacci(q*(n-j+1))/fibonacci(q*j) for j in (1..k))
%o A034802 [[F(n,k,3) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 13 2019
%o A034802 (GAP)
%o A034802 F:= function(n,k,q)
%o A034802     if n=0 and k=0 then return 1;
%o A034802     else return Product([1..k], j-> Fibonacci(q*(n-j+1))/Fibonacci(q*j));
%o A034802     fi;
%o A034802   end;
%o A034802 Flat(List([0..10], n-> List([0..n], k-> F(n,k,3) ))); # _G. C. Greubel_, Nov 13 2019
%Y A034802 Cf. A010048.
%K A034802 nonn,tabl
%O A034802 0,5
%A A034802 _N. J. A. Sloane_
%E A034802 More terms from _James Sellers_, Feb 09 2000
%E A034802 Terms of 8th row corrected by _Georg Fischer_, Dec 01 2019