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%I #35 Sep 08 2022 08:45:58
%S 1,3,24,504,27720,3991680,1504863360,1485300136320,3838015552250880,
%T 25964175210977203200,459851507161617245875200,
%U 21322394684069868456741273600,2588389457883293541569193426124800,822618641999347403739646931950148812800
%N Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).
%C The terms of this sequence are Fibonacci double factorial numbers.
%C a(n) is asymptotic to C2*phi^(n*(n+1))/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio. For the decimal expansion of C2 see A194159.
%C Product of first n terms of the binomial transform of the Fibonacci numbers. - _Vaclav Kotesovec_, Oct 29 2017
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 6th printing with corrections. Addison-Wesley, Reading, MA, p. 478 and p. 571, 1990.
%H Vincenzo Librandi, <a href="/A194157/b194157.txt">Table of n, a(n) for n = 1..70</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>, Mathworld.
%F a(n) = Product_{i=1..n} F(2*i) with F(n) = A000045(n).
%F a(n) = A123029(2*n).
%F a(n+1)/a(n) = A001906(n+1).
%F 0 = a(n)*(3*a(n+2)^2 - a(n+1)*a(n+3)) -a(n+1)^2*a(n+2) for all n>=0. - _Michael Somos_, Oct 06 2014
%p with(combinat): A194157 :=proc(n): mul(fibonacci(2*i), i=1..n) end: seq(A194157(n), n=1..14);
%t FoldList[Times, Fibonacci[2 Range[20]]] (* or *)
%t Table[Round[GoldenRatio^(n(n-1)) QFactorial[n, 1/GoldenRatio^4]], {n, 20}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)
%t Table[Product[Sum[Binomial[m, k]*Fibonacci[k], {k, 1, m}], {m, 1, n}], {n, 1, 12}] (* _Vaclav Kotesovec_, Oct 29 2017 *)
%o (PARI) {a(n) = if( n<0, 0, prod(k=1, n, fibonacci(2*k)))}; /* _Michael Somos_, Oct 06 2014 */
%o (Magma) [&*[Fibonacci(2*i): i in [1..n]]: n in [1..20]]; // _Vincenzo Librandi_, Sep 15 2016
%Y Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.
%K nonn,easy
%O 1,2
%A _Johannes W. Meijer_, Aug 21 2011
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