A186269 a(n) = Product_{k=0..n-1} A084057(k+1).

1, 1, 6, 96, 5376, 946176, 544997376, 1011515129856, 6085275021213696, 118395110812733669376, 7456050498542715562622976, 1519364146391040406489059557376, 1001953802522449942301649259468947456, 2138185445843748536070796346094885374263296, 14766000790292725890315725371457440731168428261376

Offset: 0

Author

Paul Barry, Feb 16 2011

Keywords

• nonn
• easy

Comments

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}.

Examples

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

Crossrefs

Cf. A000032, A070825, A135407, A218490.

Programs

Mathematica

Table[FullSimplify[Product[(1+Sqrt[5])^k/2 + (1-Sqrt[5])^k/2,{k,0,n}]],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)
Table[Product[LucasL[k]*2^(k-1),{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

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

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]

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