A194158 Product of first n nonzero odd-indexed Fibonacci numbers F(1), ..., F(2*n-1).

1, 2, 10, 130, 4420, 393380, 91657540, 55911099400, 89290025741800, 373321597626465800, 4086378207619294646800, 117103340295746126693347600, 8785678105688353155168403690000, 1725665322163094950031867515982420000, 887387152950606153059937200876123854180000

Offset: 1

Author

Johannes W. Meijer, Aug 21 2011

Keywords

• nonn
• easy

Comments

The terms of this sequence are Fibonacci double factorial numbers.

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.

Examples

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 130*x^4 + 4420*x^5 + 393380*x^6 + 91657540*x^7 + ...
		

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

• G. C. Greubel, Table of n, a(n) for n = 1..69
• Eric Weisstein's World of Mathematics, Fibonorial

Crossrefs

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

Programs

Maple

with(combinat): A194158 :=proc(n): mul(fibonacci(2*i-1), i=1..n) end: seq(A194158(n), n=1..15);

Mathematica

Table[Product[Fibonacci[2*k - 1], {k, 1, n}], {n, 1, 30}] (* G. C. Greubel, Aug 13 2018 *)

PARI

{a(n) = if( n<0, 1 / a(-n), prod(k=1, n, fibonacci(2*k - 1)))}; /* Michael Somos, Oct 07 2014 */

Formula

a(n) = Product_{i=1..n} F(2*i-1), where F(n) = A000045(n).

a(n) = A123029(2*n-1).

a(n+1)/a(n) = A001519(n+1).

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

a(-n) = 1/a(n) for all n in Z. - Michael Somos, Oct 07 2014

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(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