%I #30 Apr 30 2024 02:11:37

%S 1,4,30,260,2380,22428,215292,2093520,20553390,203280220,2022339176,

%T 20215564824,202879303900,2042865050800,20629119101400,

%U 208829908532880,2118554718825420,21533269718832300,219235457827640100,2235446059461106800,22824647678376163620,233331794241184490280

%N a(n) = Fibonacci(2n+1) * binomial(2n,n).

%C Central column of triangle A016095.

%C a(n) is the number of ways to tile a strip of length 2n with squares labeled 0 or 1, and dominoes labeled 00, 01, 10, or 11, where there are in total n 0's and n 1's in the tiling. - _Greg Dresden_ and _Yiming Tan_, Aug 15 2020

%F a(n) = Fibonacci(2n+1) * binomial(2n,n) = A000045(2n+1) * A000984(n). - _Philippe Deléham_, Oct 14 2006

%F a(n) = A016095(n,n).

%F Sum_{n>=0} a(n)/16^n = 2*sqrt(10+2*sqrt(5))/5. - _Amiram Eldar_, May 06 2023

%F G.f.: sqrt(3-8*x+2*sqrt(1-12*x+16*x^2))/(sqrt(5)*sqrt(1-12*x+16*x^2)). - _Vladimir Kruchinin_, Apr 30 2024

%e a(0) = F(1)*C(0,0) = 1*1 = 1;

%e a(1) = F(3)*C(2,1) = 2*2 = 4;

%e a(2) = F(5)*C(4,2) = 5*6 = 30;

%e a(3) = F(7)*C(6,3) = 13*20 = 260; ...

%t Table[Fibonacci[2n+1]Binomial[2n,n],{n,0,20}] (* _Harvey P. Dale_, Aug 03 2016 *)

%o (PARI) a(n)=fibonacci(2*n+1)*binomial(2*n,n)

%Y Cf. A000045, A000984, A016095.

%K nonn

%O 0,2

%A _Ralf Stephan_, Jan 03 2005