|
%I #31 Apr 13 2024 11:50:42
%S 1,1,1,2,48,30240,1596672000,18172937502720000,
%T 122457316443772566896640000,1284319496829094129116119090331648000000,
%U 55603466527142141932748234118927499493985767915520000000,26110840958525805673462196263372614726154694067746586937781385166848000000000
%N Vandermonde determinant of (1,2,3,...,F(n+1)), where F=A000045 (Fibonacci numbers).
%C Each term divides its successor, as in A123741. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203313.
%C For a signed version, see A123742. For a guide to related sequences, including sequences of Vandermonde permanents, see A093883.
%F 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
%e v(4) = (2-1)*(3-1)*(3-2)*(5-1)*(5-2)*(5-3).
%p with(LinearAlgebra): F:= combinat[fibonacci]:
%p a:= n-> Determinant(VandermondeMatrix([F(i)$i=2..n+1])):
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Jul 23 2017
%t f[j_] := Fibonacci[j + 1]; z = 15;
%t v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
%t d[n_] := Product[(i - 1)!, {i, 1, n}]
%t Table[v[n], {n, 1, z}] (* A203311 *)
%t Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A123741 *)
%t Table[v[n]/d[n], {n, 1, 13}] (* A203313 *)
%o (Python)
%o from sympy import fibonacci, factorial
%o from operator import mul
%o from functools import reduce
%o def f(j): return fibonacci(j + 1)
%o 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)])
%o print([v(n) for n in range(1, 16)]) # _Indranil Ghosh_, Jul 26 2017
%Y Cf. A000045, A123741, A123742, A203313, A203518.
%K nonn,changed
%O 0,4
%A _Clark Kimberling_, Jan 01 2012
| 