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