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%I #24 Sep 17 2023 01:36:45
%S 1,2,10,130,4420,393380,91657540,55911099400,89290025741800,
%T 373321597626465800,4086378207619294646800,
%U 117103340295746126693347600,8785678105688353155168403690000,1725665322163094950031867515982420000,887387152950606153059937200876123854180000
%N Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).
%C The terms of this sequence are Fibonacci double factorial numbers.
%C The a(n) is asymptotic to C1*phi^(n*n)/sqrt(5)^n where phi=(1+sqrt(5))/2 is the golden ratio A001622. For the decimal expansion of C1 see A194160.
%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 G. C. Greubel, <a href="/A194158/b194158.txt">Table of n, a(n) for n = 1..69</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>
%F a(n) = Product_{i=1..n} F(2*i-1), where F(n) = A000045(n).
%F a(n) = A123029(2*n-1).
%F a(n+1)/a(n) = A001519(n+1).
%F a(0) = 1 by convention since empty products equal 1. - _Michael Somos_, Oct 07 2014
%F a(-n) = 1/a(n) for all n in Z. - _Michael Somos_, Oct 07 2014
%F 0 = a(n)*(+a(n+1)*a(n+3) - 3*a(n+2)^2) + a(n+2)*(+a(n+1)^2) for all n in Z. - _Michael Somos_, Oct 07 2014
%F (F(1) + i)(F(3) + i)...(F(2n+1) + i) = a(n)(1 + F(2n+2)i) and (F(2n+1) + i)(1 + F(2n)i) = F(2n-1)(1 + F(2n+2)i) for all n in Z. - _Michael Somos_, Sep 16 2023
%e G.f. = 1 + x + 2*x^2 + 10*x^3 + 130*x^4 + 4420*x^5 + 393380*x^6 + 91657540*x^7 + ...
%p with(combinat): A194158 :=proc(n): mul(fibonacci(2*i-1), i=1..n) end: seq(A194158(n), n=1..15);
%t Table[Product[Fibonacci[2*k - 1], {k, 1, n}], {n, 1, 30}] (* _G. C. Greubel_, Aug 13 2018 *)
%o (PARI) {a(n) = if( n<0, 1 / a(-n), prod(k=1, n, fibonacci(2*k - 1)))}; /* _Michael Somos_, Oct 07 2014 */
%Y Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.
%K nonn,easy
%O 1,2
%A _Johannes W. Meijer_, Aug 21 2011