cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A123741 A second version of Fibonacci factorials besides A003266.

Original entry on oeis.org

1, 2, 24, 630, 52800, 11381760, 6738443712, 10487895163200, 43294107630090240, 469590163875486482400, 13388418681612808458240000, 1001088091286168023193223168000, 196239953628635168336022309340569600
Offset: 1

Views

Author

Wolfdieter Lang, Oct 13 2006

Keywords

Comments

The formula below is a generalization of n! = Product_{j=1..n} ((n+1) - j) with numbers k replaced by Fibonacci numbers F(k+1):=A000045(k+1), k>=1.
These numbers come up in Vandermonde determinants involving Fibonacci numbers [F(2),...,F(n+1)]. See A123742.

Examples

			n=3: (5-1)*(5-2)*(5-3) = 4*3*2 = 24;
n=4: (8-1)*(8-2)*(8-3)*(8-5) = 7*6*5*3 = 630.

Links

Crossrefs

Cf. A003266 (the usual Fibonacci factorials), A123742.

Programs

  • GAP
    F:=Fibonacci;; List([1..20], n-> Product([1..n], j-> F(n+2) - F(j+1))); # G. C. Greubel, Aug 10 2019
  • Magma
    F:=Fibonacci; [(&*[F(n+2)-F(j+1): j in [1..n]]): n in [1..20]] // G. C. Greubel, Aug 10 2019
  • Maple
    with(combinat): seq(mul(fibonacci(n+2)-fibonacci(j+1), j = 1..n), n = 1 .. 20); # G. C. Greubel, Aug 10 2019
  • Mathematica
    With[{F=Fibonacci}, Table[Product[F[n+2]-F[j+1],{j,n}], {n,20}]] (* G. C. Greubel, Aug 10 2019 *)
  • PARI
    vector(20, n, f=fibonacci; prod(j=1,n, f(n+2)-f(j+1))) \\ G. C. Greubel, Aug 10 2019
  • Sage
    f=fibonacci; [prod(f(n+2)-f(j+1) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = Product_{j=1..n} (F(n+2) - F(j+1)), n>=1.
a(n) ~ c * phi^(n*(n+2)) / 5^(n/2), where c = A276987 = QPochhammer(1/phi) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 31 2021

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

Original entry on oeis.org

1, 1, 1, 2, 48, 30240, 1596672000, 18172937502720000, 122457316443772566896640000, 1284319496829094129116119090331648000000, 55603466527142141932748234118927499493985767915520000000, 26110840958525805673462196263372614726154694067746586937781385166848000000000
Offset: 0

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

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.

Examples

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

Crossrefs

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

Programs

  • 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 *)
  • 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

A123743 Certain Vandermonde determinants with Fibonacci numbers.

Original entry on oeis.org

1, 2, 12, 1440, 7257600, 4981616640000, 1190690865178214400000, 272795714695463271824306995200000, 157357907118293002525216789633250308915200000000
Offset: 1

Views

Author

Wolfdieter Lang Oct 13 2006

Keywords

Comments

The determinant of a Vandermonde matrix VM_n with elements VM_n[i,j]=(x_j)^i, i,j,=1..n, is VdmII([x_1,...,x_n]) := Det(VM_n)= product(x_k,k=1,...,n)*product(x_j - x_i, 1<=i=2. For n=1, Det(VM_1)=1.

Examples

			n=4: VM_4 = matrix([1,2,3,5],[1,4,9,25],[1,8,27,125],[1,16,81,625]).
a(4)=Det(VM_4) = 1440 = 30*48 = A003266(5)*|A123742(4)|.

Crossrefs

Cf. A123742 (another version).

Formula

a(n)= Fibfac(n)* |A123742(n)|, with the Fibonacci factorials Fibfac(n):=A003266(n+1).
a(n)=VdmII([F(2),F(3),...,F(n+1)]) := Det(VM_n[i,j]) with the Vandermonde matrix elements VM_n[i,j]:=F(j+1)^i, i,j,=1..n and F(k):=A000045(k) (Fibonacci).
Showing 1-3 of 3 results.

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