A194157 Product of first n nonzero even-indexed Fibonacci numbers F(2), F(4), F(6), ..., F(2*n).

1, 3, 24, 504, 27720, 3991680, 1504863360, 1485300136320, 3838015552250880, 25964175210977203200, 459851507161617245875200, 21322394684069868456741273600, 2588389457883293541569193426124800, 822618641999347403739646931950148812800

Offset: 1

Author

Johannes W. Meijer, Aug 21 2011

Keywords

• nonn
• easy

Comments

The terms of this sequence are Fibonacci double factorial numbers.

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.

Product of first n terms of the binomial transform of the Fibonacci numbers. - Vaclav Kotesovec, Oct 29 2017

References

• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..70
• Eric Weisstein, Fibonorial, Mathworld.

Crossrefs

Cf. A003266, A062073, A194158, A194160, A194157, A194159, A123029.

Programs

Magma

[&*[Fibonacci(2*i): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Sep 15 2016

Maple

with(combinat): A194157 :=proc(n): mul(fibonacci(2*i), i=1..n) end: seq(A194157(n), n=1..14);

Mathematica

FoldList[Times, Fibonacci[2 Range[20]]] (* or *)
Table[Round[GoldenRatio^(n(n-1)) QFactorial[n, 1/GoldenRatio^4]], {n, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
Table[Product[Sum[Binomial[m, k]*Fibonacci[k], {k, 1, m}], {m, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Oct 29 2017 *)

PARI

{a(n) = if( n<0, 0, prod(k=1, n, fibonacci(2*k)))}; /* Michael Somos, Oct 06 2014 */

Formula

a(n) = Product_{i=1..n} F(2*i) with F(n) = A000045(n).

a(n) = A123029(2*n).

a(n+1)/a(n) = A001906(n+1).

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