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A123741 A second version of Fibonacci factorials besides A003266.

%I A123741 #16 Sep 08 2022 08:45:28
%S A123741 1,2,24,630,52800,11381760,6738443712,10487895163200,
%T A123741 43294107630090240,469590163875486482400,13388418681612808458240000,
%U A123741 1001088091286168023193223168000,196239953628635168336022309340569600
%N A123741 A second version of Fibonacci factorials besides A003266.
%C A123741 The formula below is a generalization of n! = Product_{j=1..n} ((n+1) - j) with numbers k replaced by Fibonacci numbers F(k+1):=A000045(k+1), k>=1.
%C A123741 These numbers come up in Vandermonde determinants involving Fibonacci numbers [F(2),...,F(n+1)]. See A123742.
%H A123741 G. C. Greubel, <a href="/A123741/b123741.txt">Table of n, a(n) for n = 1..68</a>
%F A123741 a(n) = Product_{j=1..n} (F(n+2) - F(j+1)), n>=1.
%F A123741 a(n) ~ c * phi^(n*(n+2)) / 5^(n/2), where c = A276987 = QPochhammer(1/phi) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 31 2021
%e A123741 n=3: (5-1)*(5-2)*(5-3) = 4*3*2 = 24;
%e A123741 n=4: (8-1)*(8-2)*(8-3)*(8-5) = 7*6*5*3 = 630.
%p A123741 with(combinat): seq(mul(fibonacci(n+2)-fibonacci(j+1), j = 1..n), n = 1 .. 20); # _G. C. Greubel_, Aug 10 2019
%t A123741 With[{F=Fibonacci}, Table[Product[F[n+2]-F[j+1],{j,n}], {n,20}]] (* _G. C. Greubel_, Aug 10 2019 *)
%o A123741 (PARI) vector(20, n, f=fibonacci; prod(j=1,n, f(n+2)-f(j+1))) \\ _G. C. Greubel_, Aug 10 2019
%o A123741 (Magma) F:=Fibonacci; [(&*[F(n+2)-F(j+1): j in [1..n]]): n in [1..20]] // _G. C. Greubel_, Aug 10 2019
%o A123741 (Sage) f=fibonacci; [prod(f(n+2)-f(j+1) for j in (1..n)) for n in (1..20)] # _G. C. Greubel_, Aug 10 2019
%o A123741 (GAP) F:=Fibonacci;; List([1..20], n-> Product([1..n], j-> F(n+2) - F(j+1))); # _G. C. Greubel_, Aug 10 2019
%Y A123741 Cf. A003266 (the usual Fibonacci factorials), A123742.
%K A123741 nonn,easy
%O A123741 1,2
%A A123741 _Wolfdieter Lang_, Oct 13 2006