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A034801 Triangle of Fibonomial coefficients (k=2).

%I A034801 #18 Jul 02 2025 16:01:56
%S A034801 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,
%T A034801 2640,144,1,1,377,18096,124410,124410,18096,377,1,1,987,124033,
%U A034801 2232594,5847270,2232594,124033,987,1,1,2584,850136,40062659,274715376,274715376,40062659,850136,2584,1
%N A034801 Triangle of Fibonomial coefficients (k=2).
%D A034801 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 88.
%H A034801 G. C. Greubel, <a href="/A034801/b034801.txt">Rows n = 0..100 of triangle, flattened</a>
%H A034801 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.
%H A034801 C. J. Pita Ruiz Velasco, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita2/pita8.html">Sums of Products of s-Fibonacci Polynomial Sequences</a>, J. Int. Seq. 14 (2011) # 11.7.6.
%F A034801 Fibonomial coefficients formed from sequence F_3k [ 2, 8, 34, ... ].
%F A034801 T(n, k) = Product_{j=0..k-1} Fibonacci(2*(n-j)) / Product_{j=1..k} Fibonacci(2*j).
%e A034801 Triangle begins as:
%e A034801   1;
%e A034801   1,   1;
%e A034801   1,   3,      1;
%e A034801   1,   8,      8,       1;
%e A034801   1,  21,     56,      21,       1;
%e A034801   1,  55,    385,     385,      55,       1;
%e A034801   1, 144,   2640,    6930,    2640,     144,      1;
%e A034801   1, 377,  18096,  124410,  124410,   18096,    377,   1;
%e A034801   1, 987, 124033, 2232594, 5847270, 2232594, 124033, 987, 1;
%p A034801 A034801 := proc(n,k)
%p A034801     mul(combinat[fibonacci](2*n-2*j),j=0..k-1) /
%p A034801     mul(combinat[fibonacci](2*j),j=1..k) ;
%p A034801 end proc: # _R. J. Mathar_, Sep 02 2017
%t A034801 F[n_, k_, q_]:= Product[Fibonacci[q*(n-j+1)]/Fibonacci[q*j], {j,k}];
%t A034801 Table[F[n, k, 2], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 13 2019 *)
%o A034801 (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 A034801 (Sage)
%o A034801 def F(n,k,q):
%o A034801     if (n==0 and k==0): return 1
%o A034801     else: return product(fibonacci(q*(n-j+1))/fibonacci(q*j) for j in (1..k))
%o A034801 [[F(n,k,2) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 13 2019
%o A034801 (GAP)
%o A034801 F:= function(n,k,q)
%o A034801     if n=0 and k=0 then return 1;
%o A034801     else return Product([1..k], j-> Fibonacci(q*(n-j+1))/Fibonacci(q*j));
%o A034801     fi;
%o A034801   end;
%o A034801 Flat(List([0..10], n-> List([0..n], k-> F(n,k,2) ))); # _G. C. Greubel_, Nov 13 2019
%Y A034801 Cf. A010048.
%K A034801 nonn,tabl
%O A034801 0,5
%A A034801 _N. J. A. Sloane_
%E A034801 More terms from _James Sellers_, Feb 09 2000