proposed

approved

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proposed

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.

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.

proposed

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proposed

[[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)

(GAP)

F:= function(n, k, q)

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

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

aT(n, k) = product(fibonacci(2*(n-j)), Product_{j=0..k-1)/product(fibonacci} Fibonacci(2*(n-j), ) / Product_{j=1..k} Fibonacci(2*j).

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 ;

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(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

N. J. A. Sloane.

approved

editing

editing

approved

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

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 ;

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

approved

editing

editing

approved

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

approved

editing