A123029 a(2*n-1) = Product_{i=1..n} Fibonacci(2*i-1) and a(2*n) = Product_{i=1..n} Fibonacci(2*i).

1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540, 1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880, 373321597626465800, 25964175210977203200, 4086378207619294646800

Offset

1,3

Comments

From Johannes W. Meijer, Aug 21 2011: (Start)

An appropriate name for this sequence is Fibonacci double factorial, cf. A006882.

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)

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)).

Bagula also stated that using the solutions to these second order differential equations Markov/ linear recursions can be encoded as analog functions.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..100

P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov, Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.

Eric Weisstein's World of Mathematics, Fibonorial.

Formula

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).

From Johannes W. Meijer, Aug 21 2011: (Start)

a(n) = F(n)*a(n-2).

a(2*n) = A194157(n) and a(2*n-1) = A194158(n). (End)

a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 07 2014

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

Example

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 10*x^5 + 24*x^6 + 130*x^7 + 504*x^8 + ...

Maple

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

Mathematica

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 *)
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 *)

Prog

(PARI) {a(n) = if( n<0, 0, prod(k=0, (n-1)\2, fibonacci(n - 2*k)))}; /* Michael Somos, Oct 07 2014 */
(Magma)
function a(n)
  if n lt 2 then return 1;
  else return Fibonacci(n)*a(n-2);
  end if; return a;
end function;
[a(n): n in [1..30]]; // G. C. Greubel, Jul 20 2021
(Sage)
def a(n): return 1 if (n<2) else fibonacci(n)*a(n-2)
[a(n) for n in (1..30)] # G. C. Greubel, Jul 20 2021

Crossrefs

Cf. A000045, A003266, A194157, A194158.

Cf. A006882, A103631, A194005.

Sequence in context: A105286 A295616 A059929 * A103018 A246437 A341265

Adjacent sequences: A123026 A123027 A123028 * A123030 A123031 A123032

Keyword

nonn,easy

Author

Roger L. Bagula, Sep 25 2006

Extensions

Edited by Johannes W. Meijer, Aug 21 2011

Status

approved