A203311 Vandermonde determinant of (1,2,3,...,F(n+1)), where F=A000045 (Fibonacci numbers).

1, 1, 1, 2, 48, 30240, 1596672000, 18172937502720000, 122457316443772566896640000, 1284319496829094129116119090331648000000, 55603466527142141932748234118927499493985767915520000000, 26110840958525805673462196263372614726154694067746586937781385166848000000000

Offset

0,4

Comments

Each term divides its successor, as in A123741. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203313.

For a signed version, see A123742. For a guide to related sequences, including sequences of Vandermonde permanents, see A093883.

Links

Table of n, a(n) for n=0..11.

Formula

a(n) ~ c * d^n * phi^(n^3/3 + n^2/2) / 5^(n^2/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio, d = 0.120965069090607877853843907542896935455225485213927649233956250456604334... and c = 197.96410442333389877538426269... - Vaclav Kotesovec, Apr 08 2021

Example

v(4) = (2-1)*(3-1)*(3-2)*(5-1)*(5-2)*(5-3).

Maple

with(LinearAlgebra): F:= combinat[fibonacci]:
a:= n-> Determinant(VandermondeMatrix([F(i)$i=2..n+1])):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

Mathematica

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]
Table[v[n], {n, 1, z}]                (* A203311 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A123741 *)
Table[v[n]/d[n], {n, 1, 13}]          (* A203313 *)

Prog

(Python)
from sympy import fibonacci, factorial
from operator import mul
from functools import reduce
def f(j): return fibonacci(j + 1)
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [f(k) - f(j) for j in range(1, k)]) for k in range(2, n + 1)])
print([v(n) for n in range(1, 16)]) # Indranil Ghosh, Jul 26 2017

Crossrefs

Cf. A000045, A123741, A123742, A203313, A203518.

Sequence in context: A090770 A081960 A123742 * A343694 A295177 A374928

Adjacent sequences: A203308 A203309 A203310 * A203312 A203313 A203314

Keyword

nonn

Author

Clark Kimberling, Jan 01 2012

Status

approved