A082471 - OEIS

%I #38 Apr 07 2020 03:34:04

%S 1,1,2,6,24,144,1296,18144,399168,13970880,782369280,70413235200,

%T 10209919104000,2389121070336000,903087764587008000,

%U 551786624162661888000,545165184672709945344000,871173965106990492659712000,2251984699801570423525355520000

%N a(1)=1, a(n) = Sum_{k=1..n-1} Fibonacci(k)*a(k).

%H Vincenzo Librandi, <a href="/A082471/b082471.txt">Table of n, a(n) for n = 1..100</a>

%H Thotsaporn Aek Thanatipanonda and Yi Zhang, <a href="https://arxiv.org/abs/2004.01370">Sequences: Polynomial, C-finite, Holonomic, ...</a>, arXiv:2004.01370 [math.CO], 2020.

%F For n >= 2, a(n) = (F(n-1) + 1)*a(n-1); a(n) = (1/2)*Product_{k=1..n-1} (F(k)+1).

%F G.f.: 1 + x/(G(0)-2*x) where G(k) = 1 + x + x*F(k+1) - x*(F(k+2)+1)/G(k+1); F(k) is the k-th Fibonacci number; (continued fraction). - _Sergei N. Gladkovskii_, Jul 08 2012

%F a(n) ~ c * ((1+sqrt(5))/2)^(n*(n-1)/2) / 5^(n/2), where c = 18.0370671229828603013612398720270653807943654417062957419698762672485... - _Vaclav Kotesovec_, Aug 14 2017

%t Join[{1}, 1/2 Table[Product[Fibonacci[k] + 1, {k, 1, n}], {n, 1, 20}]] (* _Vincenzo Librandi_, Aug 14 2017 *)

%o (PARI) a(n) = if (n==1, 1, prod(k=1, n-1, fibonacci(k)+1)/2); \\ _Michel Marcus_, Aug 14 2017

%Y Cf. A003266.

%K nonn

%O 1,3

%A _Benoit Cloitre_, Apr 27 2003

%E More terms from _Michel Marcus_, Aug 14 2017