%I #8 Jul 11 2015 08:21:59

%S 1,1,3,21,357,14637,1449063,346326057,199830134889,278363377900377,

%T 936136039878967851,7600488507777339982269,

%U 148977175240943640992454669,7049748909576694035403947391749,805384464676770256686653161875581007

%N a(n)=Product{k=0..n, A001333(k)}.

%C a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), Pell(j+1),

%C Pell(i+1)))_{0<=i,j<=n}, where Pell(n)=A000129(n).

%F a(n)=Product{k=0..n, sum{j=0..floor(k/2), binomial(k,2j)*2^j}}.

%F a(n) ~ c * (1+sqrt(2))^(n*(n+1)/2) / 2^(n+1), where c = 1.6982679851338713863950411843311686297311132648098280324748781109134... . - _Vaclav Kotesovec_, Jul 11 2015

%e a(3)=21 since det[1, 1, 1, 1; 1, 2, 2, 2; 1, 2, 5, 5; 1, 2, 5, 12]=21.

%t Table[Product[Sum[Binomial[k,2*j]*2^j,{j,0,Floor[k/2]}],{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Jul 11 2015 *)

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

%Y Cf. A186269.

%K nonn,easy

%O 0,3

%A _Paul Barry_, Feb 16 2011