A186269 - OEIS

%I #21 Feb 12 2025 03:14:22

%S 1,1,6,96,5376,946176,544997376,1011515129856,6085275021213696,

%T 118395110812733669376,7456050498542715562622976,

%U 1519364146391040406489059557376,1001953802522449942301649259468947456,2138185445843748536070796346094885374263296,14766000790292725890315725371457440731168428261376

%N a(n) = Product_{k=0..n-1} A084057(k+1).

%C a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), 2^j*F(j+1), 2^i*F(i+1)))_{0<=i,j<=n}.

%F a(n) = Product_{k=0..n} (1+sqrt(5))^k/2+(1-sqrt(5))^k/2.

%F a(n) = Product_{k=0..n} Sum_{j=0..floor(k/2)} binomial(k,2*j)*5^j. [corrected by _Jason Yuen_, Feb 12 2025]

%F a(n) ~ c * (1+sqrt(5))^(n*(n+1)/2) / 2^(n+1), where c = A218490 = 1.3578784076121057013874397... is the Lucas factorial constant. - _Vaclav Kotesovec_, Jul 11 2015

%e a(2)=6 since det[1, 1, 1; 1, 2, 2; 1, 2, 8]=6.

%t Table[FullSimplify[Product[(1+Sqrt[5])^k/2 + (1-Sqrt[5])^k/2,{k,0,n}]],{n,0,15}] (* _Vaclav Kotesovec_, Jul 11 2015 *)

%t Table[Product[LucasL[k]*2^(k-1),{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Jul 11 2015 *)

%Y Cf. A000032, A070825, A135407, A218490.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Feb 16 2011