%I A068875
%S A068875 1,2,4,10,28,84,264,858,2860,9724,33592,117572,416024,1485800,5348880,
%T A068875 19389690,70715340,259289580,955277400,3534526380,13128240840,
%U A068875 48932534040,182965127280,686119227300,2579808294648,9723892802904
%N A068875 Expansion of (1+x*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for 
               Catalan numbers, A000108.
%C A068875 A Catalan transform of A040000 under the mapping g(x)->g(xc(x)). A040000 
               can be retrieved using the mapping g(x)->g(x(1-x)). A040000(n)=sum{k=0..floor(n/
               2), C(n-k,k)(-1)^k*A068875(n-k)}. A068875 and A040000 may be described 
               as a Catalan pair. - Paul Barry (pbarry(AT)wit.ie), Nov 14 2004
%C A068875 a(n) = number of Dyck (n+1)-paths all of whose nonterminal descents to 
               ground level are of odd length. For example, a(2) counts UUUDDD, 
               UUDUDD, UDUUDD, UDUDUD. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 25 2005
%D A068875 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%F A068875 Apart from initial term, twice Catalan numbers.
%F A068875 G.f.: (1+xc(x))/(1-xc(x)), where c(x) is the g.f. of A000108; a(n)=C(n)(2-0^n); 
               C(n) as in A000108; a(n)=sum{j=0..n, sum{k=0..n, C(2n, n-k)((2k+1)/
               (n+k+1))C(k, j)(-1)^(j-k)*(2-0^j)}}. - Paul Barry (pbarry(AT)wit.ie), 
               Nov 14 2004
%F A068875 Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael 
               Somos Aug 17 2005
%F A068875 Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so 
               A(x)=C(x)^2-C(x)^4, A(A(x))=C(x)^4-C(x)^8, A(A(A(x)))=C(x)^8-C(x)^16, 
               etc., where C(x)=(1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 
               +... - Paul D. Hanna (pauldhanna(AT)juno.com), May 16 2008
%p A068875 Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): 
               seq(coeff(Gser, x, n), n=2..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 23 2006
%p A068875 Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser*4^n, 
               z, n), n=1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 01 2007
%o A068875 (PARI) a(n)=if(n<1, n==0, 2*binomial(2*n,n)/(n+1)) /* Michael Somos Aug 
               17 2005 */
%o A068875 (PARI) a(n)=if(n<1, n==0, polcoeff(4/(1+sqrt(1-4*x+x*O(x^n))),n)) /* 
               Michael Somos Aug 17 2005 */
%Y A068875 Cf. A068875, A068875.
%Y A068875 Sequence in context: A148110 A149823 A149824 this_sequence A135336 A149825 
               A149826
%Y A068875 Adjacent sequences: A068872 A068873 A068874 this_sequence A068876 A068877 
               A068878
%K A068875 nonn
%O A068875 0,2
%A A068875 N. J. A. Sloane (njas(AT)research.att.com), Jun 06 2002