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A123029 a(2*n-1) = Product_{i=1..n} fibonacci(2*i-1) and a(2*n) = Product_{i=1..n} fibonacci(2*i). 4
1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540, 1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880, 373321597626465800, 25964175210977203200, 4086378207619294646800 (list; graph; refs; listen; history; text; internal format)
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 "A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov" in 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!*a1(n) with a1(n) = b(n)*a1(n-2)/(n*(n-1)), a1(0) = 1, a1(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(n) = 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

f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == a[n - 1] + a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] ; Clear[a] a[n_] := a[n] = f[n]*a[n - 2]/(n*(n - 1)); a[0] = 1; a[1] = 1; Table[ExpandAll[a[n]*n! ], {n, 0, 30}]

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 */

CROSSREFS

Cf. A000045, A003266, A194157 and A194158.

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 and information added by Johannes W. Meijer, Aug 21 2011

STATUS

approved

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Last modified April 15 20:40 EDT 2021. Contains 342977 sequences. (Running on oeis4.)