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%I #41 Sep 16 2023 17:13:25
%S 1,1,2,3,10,24,130,504,4420,27720,393380,3991680,91657540,1504863360,
%T 55911099400,1485300136320,89290025741800,3838015552250880,
%U 373321597626465800,25964175210977203200,4086378207619294646800
%N a(2*n-1) = Product_{i=1..n} Fibonacci(2*i-1) and a(2*n) = Product_{i=1..n} Fibonacci(2*i).
%C From _Johannes W. Meijer_, Aug 21 2011: (Start)
%C An appropriate name for this sequence is Fibonacci double factorial, cf. A006882.
%C In Parks' article appendix 2, a number triangle T(n,k) with T(n,n) = a(n+1), n>=0, appears if we assume that b(r) = Fibonacci(r); see A103631 and A194005. (End)
%C The original name of this sequence was: A000045 inside a second linear differential equation recursion: b(n) = b(n-1) + b(n-2) --> Binet(n) of A000045 a(n) = b(n)*a(n-2)/(n*(n-1)).
%C Bagula also stated that using the solutions to these second order differential equations Markov/ linear recursions can be encoded as analog functions.
%C Partial products of the odd-indexed Fibonacci numbers interleaved with the partial products of the even-indexed Fibonacci numbers. - _Harvey P. Dale_, Mar 14 2012
%H Vincenzo Librandi, <a href="/A123029/b123029.txt">Table of n, a(n) for n = 1..100</a>
%H P. C. Parks, <a href="https://doi.org/10.1017/S030500410004072X"> A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov</a>, Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>.
%F a(n) = n!*c(n) with c(n) = b(n)*c(n-2)/(n*(n-1)), c(0) = 1, c(1) = 1; b(n) = b(n-1) + b(n-2), b(0) = 0, b(1) = 1 and b(n) = F(n) with F(n) = A000045(n).
%F From _Johannes W. Meijer_, Aug 21 2011: (Start)
%F a(n) = F(n)*a(n-2).
%F a(2*n) = A194157(n) and a(2*n-1) = A194158(n). (End)
%F a(0) = 1 by convention since empty products equal 1. - _Michael Somos_, Oct 07 2014
%F 0 = a(n)*(a(n+2)*a(n+3) - a(n+1)*a(n+4)) + a(n+1)*(+a(n+2)^2) for all n>=0. - _Michael Somos_, Oct 07 2014
%e G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 24*x^6 + 130*x^7 + 504*x^8 + ...
%p with(combinat): A123029 :=proc(n): if type(n,even) then mul(fibonacci(2*i), i=1..n/2) else mul(fibonacci(2*i-1), i= 1..(n+1)/2) fi: end: seq(A123029(n), n=1..21); # _Johannes W. Meijer_, Aug 21 2011
%t a[n_]:= a[n]= If[n<2, 1, Fibonacci[n]*a[n-2]]; Table[a[n], {n, 30}] (* modified by _G. C. Greubel_, Jul 20 2021 *)
%t With[{nn=21},Riffle[FoldList[Times,1,Fibonacci[Range[3,nn,2]]],FoldList[ Times,1, Fibonacci[ Range[4,nn+1,2]]]]] (* _Harvey P. Dale_, Mar 14 2012 *)
%o (PARI) {a(n) = if( n<0, 0, prod(k=0, (n-1)\2, fibonacci(n - 2*k)))}; /* _Michael Somos_, Oct 07 2014 */
%o (Magma)
%o function a(n)
%o if n lt 2 then return 1;
%o else return Fibonacci(n)*a(n-2);
%o end if; return a;
%o end function;
%o [a(n): n in [1..30]]; // _G. C. Greubel_, Jul 20 2021
%o (Sage)
%o def a(n): return 1 if (n<2) else fibonacci(n)*a(n-2)
%o [a(n) for n in (1..30)] # _G. C. Greubel_, Jul 20 2021
%Y Cf. A000045, A003266, A194157, A194158.
%Y Cf. A006882, A103631, A194005.
%K nonn,easy
%O 1,3
%A _Roger L. Bagula_, Sep 25 2006
%E Edited by _Johannes W. Meijer_, Aug 21 2011
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