%I A085139
%S A085139 1,1,2,6,18,58,194,670,2370,8546,31298,116102,435346,1647418,6283394,
%T A085139 24130174,93226242,362098050,1413098370,5538138182,21788069266,
%U A085139 86016385274,340655956802,1353023683486,5388230857538,21510345134178
%N A085139 a(0)=1, for n>0: a(n)=(1/2) Sum( Sum( a(j)a(i-j), (j=0..i)) (1+(-1)^(n+1+i)), 
               (i=0..n)).
%F A085139 G.f.: A(x)=(1/(2x))(1 - x^2 - Sqrt[(1 - x^2)^2 - 4x(1 - x^2)])
%F A085139 G.f.: c(x/(1-x^2)) where c(x) is the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), 
               Apr 12 2005
%F A085139 a(n)=sum{k=0..floor(n/2), C(n-k,k)C(n-2k)}-sum{k=0..,floor((n-2)/2), 
               C(n-k-2,k)C(n-2k-2)}; [From Paul Barry (pbarry(AT)wit.ie), Nov 30 
               2008]
%F A085139 Contribution from Paul Barry (pbarry(AT)wit.ie), May 27 2009: (Start)
%F A085139 G.f.: 1+x/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/
               (1-2x-2x^2/(1-x-2x^2/(1-... (continued fraction).
%F A085139 a(n)=0^n+sum{k=0..floor((n-1)/2), C(n-k-1,k)*A000108(n-2k)}. (End)
%t A085139 a[n_] := a[n] = (1/2)Sum[Sum[a[j]a[i -j], {j, 0, i}](1 + (-1)^(n+1+i)), 
               {i, 0, n}]; a[0] = 1; Table[a[n], {n, 0, 10}]
%Y A085139 Sequence in context: A000137 A151282 A157004 this_sequence A150041 A150042 
               A036675
%Y A085139 Adjacent sequences: A085136 A085137 A085138 this_sequence A085140 A085141 
               A085142
%K A085139 easy,nonn
%O A085139 0,3
%A A085139 Mario Catalani (mario.catalani(AT)unito.it), Jun 20 2003