A203520 -id:A203520 - cp's OEIS frontend

A203518 a(n) = Product_{2 <= i < j <= n+1} (F(i) + F(j)), where F = A000045 (Fibonacci numbers).

1, 3, 60, 20160, 259459200, 329533940736000, 102591687479575117824000, 20251578856869019790329341542400000, 6518596139761671764183992268499872995344384000000, 8899914870403074273776879003081000194727401271025610417766400000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203519. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n). See A093883 for a guide to related sequences.

Cf. A000045, A203519.

F:= combinat[fibonacci]:
a:= n-> mul(mul(F(i)+F(j), i=2..j-1), j=3..n+1):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

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}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

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 = 3.99077126463315610748163699882855013294148355045548571306491607634698645935... and c = 0.019290318831631524125422284... - Vaclav Kotesovec, Apr 09 2021

Name edited by Alois P. Heinz, Jul 23 2017

A203519 a(n) = v(n+1)/v(n), where v=A203518.

Original entry on oeis.org

3, 20, 336, 12870, 1270080, 311323584, 197399802600, 321880885724160, 1365311591573529600, 15068868587132753685600, 434169705562891299584593920, 32678748925653999616045678080000, 6431834564578466234122576826339121600

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Cf. A203518, A000045, A093883.

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}] (* A000178(n-1) *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

a(n) ~ c * phi^(n*(n+2) + 5/6) / 5^(n/2 + 1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 3.990771264633156107481636998828550132941483550455485713064916076346986459357... - Vaclav Kotesovec, Apr 09 2021

cp's OEIS Frontend

A203518 a(n) = Product_{2 <= i < j <= n+1} (F(i) + F(j)), where F = A000045 (Fibonacci numbers).

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