%I A098597
%S A098597 1,1,1,5,7,21,33,429,715,2431,4199,29393,52003,185725,334305,9694845,
%T A098597 17678835,64822395,119409675,883631595,1641030105,6116566755,
%U A098597 11435320455,171529806825,322476036831,1215486600363,2295919134019
%N A098597 Numerator of Catalan(n)/2^(2n+1). Also, numerators of (2n-1)!!/(n+1)!. 
               Odd part of the n-th Catalan number.
%C A098597 Also numerators of g.f. c(x/2)=(1-sqrt(1-2x))/x where c(x)=g.f. of A000108. 
               - Paul Barry (pbarry(AT)wit.ie), Sep 04 2007
%C A098597 Also numerator of x(n)=Sum(x(k)*x(n-k-1):0<=k<n), x(0)=1/2: x(n)=a(n)/
               A086117(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 06 2008
%F A098597 Numerators of g.f.: 1/(1+sqrt(1-x)).
%F A098597 a(n) = A000108(n) / 2^A048881(n).
%e A098597 1/(1+sqrt(1-x)) = 1/2 + 1/8*x + 1/16*x^2 + 5/128*x^3 + 7/256*x^4 +...
%p A098597 Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), 
               j=1..30): gser:=series(g, z=0, 27): seq(numer(coeff(gser, z, n)), 
               n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 
               2008
%p A098597 a:= n-> abs (numer (binomial (1/2, n+1))): seq (a(n), n=0..50); [From 
               Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 10 2009]
%o A098597 (PARI) a(n)=if(n<0,0,numerator(polcoeff(1/(1+sqrt(1-x+x^n*O(x))),n)))
%Y A098597 Cf. Equals A000265(A000108(n)).
%Y A098597 Essentially the absolute values of A002596. Cf. A000108, A001795.
%Y A098597 Sequence in context: A027152 A076197 A002596 this_sequence A097038 A049114 
               A030735
%Y A098597 Adjacent sequences: A098594 A098595 A098596 this_sequence A098598 A098599 
               A098600
%K A098597 nonn,frac
%O A098597 0,4
%A A098597 Michael Somos, Sep 15 2004
%E A098597 Edited by Ralf Stephan, Dec 28 2004