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

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