A123741 A second version of Fibonacci factorials besides A003266.

1, 2, 24, 630, 52800, 11381760, 6738443712, 10487895163200, 43294107630090240, 469590163875486482400, 13388418681612808458240000, 1001088091286168023193223168000, 196239953628635168336022309340569600

Offset: 1

Author

Wolfdieter Lang, Oct 13 2006

Keywords

• nonn
• easy

Comments

The formula below is a generalization of n! = Product_{j=1..n} ((n+1) - j) with numbers k replaced by Fibonacci numbers F(k+1):=A000045(k+1), k>=1.

These numbers come up in Vandermonde determinants involving Fibonacci numbers [F(2),...,F(n+1)]. See A123742.

Examples

			n=3: (5-1)*(5-2)*(5-3) = 4*3*2 = 24;
n=4: (8-1)*(8-2)*(8-3)*(8-5) = 7*6*5*3 = 630.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..68

Crossrefs

Cf. A003266 (the usual Fibonacci factorials), A123742.

Programs

GAP

F:=Fibonacci;; List([1..20], n-> Product([1..n], j-> F(n+2) - F(j+1))); # G. C. Greubel, Aug 10 2019

Magma

F:=Fibonacci; [(&*[F(n+2)-F(j+1): j in [1..n]]): n in [1..20]] // G. C. Greubel, Aug 10 2019

Maple

with(combinat): seq(mul(fibonacci(n+2)-fibonacci(j+1), j = 1..n), n = 1 .. 20); # G. C. Greubel, Aug 10 2019

Mathematica

With[{F=Fibonacci}, Table[Product[F[n+2]-F[j+1],{j,n}], {n,20}]] (* G. C. Greubel, Aug 10 2019 *)

PARI

vector(20, n, f=fibonacci; prod(j=1,n, f(n+2)-f(j+1))) \\ G. C. Greubel, Aug 10 2019

Sage

f=fibonacci; [prod(f(n+2)-f(j+1) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = Product_{j=1..n} (F(n+2) - F(j+1)), n>=1.

a(n) ~ c * phi^(n*(n+2)) / 5^(n/2), where c = A276987 = QPochhammer(1/phi) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 31 2021