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Triangle of Fibonomial coefficients (k=2).
8

%I #17 Nov 13 2019 15:01:17

%S 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 2640,144,1,1,377,18096,124410,124410,18096,377,1,1,987,124033,

%U 2232594,5847270,2232594,124033,987,1,1,2584,850136,40062659,274715376,274715376,40062659,850136,2584,1

%N Triangle of Fibonomial coefficients (k=2).

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

%H G. C. Greubel, <a href="/A034801/b034801.txt">Rows n = 0..100 of triangle, flattened</a>

%H 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 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 Fibonomial coefficients formed from sequence F_3k [ 2, 8, 34, ... ].

%F T(n, k) = Product_{j=0..k-1} Fibonacci(2*(n-j)) / Product_{j=1..k} Fibonacci(2*j).

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 8, 8, 1;

%e 1, 21, 56, 21, 1;

%e 1, 55, 385, 385, 55, 1;

%e 1, 144, 2640, 6930, 2640, 144, 1;

%e 1, 377, 18096, 124410, 124410, 18096, 377, 1;

%e 1, 987, 124033, 2232594, 5847270, 2232594, 124033, 987, 1;

%p A034801 := proc(n,k)

%p mul(combinat[fibonacci](2*n-2*j),j=0..k-1) /

%p mul(combinat[fibonacci](2*j),j=1..k) ;

%p end proc: # _R. J. Mathar_, Sep 02 2017

%t F[n_, k_, q_]:= Product[Fibonacci[q*(n-j+1)]/Fibonacci[q*j], {j,k}];

%t Table[F[n, k, 2], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 13 2019 *)

%o (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 (Sage)

%o def F(n,k,q):

%o if (n==0 and k==0): return 1

%o else: return product(fibonacci(q*(n-j+1))/fibonacci(q*j) for j in (1..k))

%o [[F(n,k,2) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 13 2019

%o (GAP)

%o F:= function(n,k,q)

%o if n=0 and k=0 then return 1;

%o else return Product([1..k], j-> Fibonacci(q*(n-j+1))/Fibonacci(q*j));

%o fi;

%o end;

%o Flat(List([0..10], n-> List([0..n], k-> F(n,k,2) ))); # _G. C. Greubel_, Nov 13 2019

%Y Cf. A010048.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Feb 09 2000