A302999 a(n) = Product_{k=1..n} (Fibonacci(k+2) - 1).

1, 1, 2, 8, 56, 672, 13440, 443520, 23950080, 2107607040, 301387806720, 69921971159040, 26290661155799040, 16011012643881615360, 15786858466867272744960, 25195826113120167300956160, 65080818850189392138369761280, 272037822793791659138385602150400

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

• nonn

Comments

a(n) = determinant of (n + 1) X (n + 1) matrix whose main diagonal consists of the consecutive Fibonacci numbers starting with Fibonacci(2) (1, 2, 3, 5, 8, 13, ...) and all other elements are 1's (see example).

Examples

			The matrix begins:
  1  1  1  1  1  1  1  1  ...
  1  2  1  1  1  1  1  1  ...
  1  1  3  1  1  1  1  1  ...
  1  1  1  5  1  1  1  1  ...
  1  1  1  1  8  1  1  1  ...
  1  1  1  1  1 13  1  1  ...
  1  1  1  1  1  1 21  1  ...
  1  1  1  1  1  1  1 34  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..97

Crossrefs

Cf. A000045, A000071, A003266, A067962, A190535, A194157, A194158, A196870.

Programs

Maple

b:= proc(n) b(n):= `if`(n<1, [1$2][], (f->
      [f, b(n-1)[2]*(f-1)][])(b(n-1)+b(n-2)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 24 2018

Mathematica

Table[Product[Fibonacci[k + 2] - 1, {k, 1, n}], {n, 0, 17}]
Table[Product[Sum[Fibonacci[j], {j, 1, k}], {k, 1, n}], {n, 0, 17}]
Table[Det[Table[If[i == j, Fibonacci[i + 1], 1], {i, 1, n + 1}, {j, 1, n + 1}]], {n, 0, 17}]

Formula

a(n) = Product_{k=1..n} A000071(k+2).

a(n) = Product_{k=1..n} Sum_{j=1..k} A000045(j).

a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+5)/2) / 5^(n/2), where c = 0.1972502311584232476952451740107000852343536766534965116633336539193... - Vaclav Kotesovec, Apr 17 2018

a(n) = A190535(n-3) for n > 3. - Alois P. Heinz, Apr 25 2018