%I
%S 0,1,2,7,21,69,228,773,2659,9275,32715,116511,418377,1513163,5507242,
%T 20155583,74131537,273862373,1015762117,3781095113,14121051487,
%U 52895245133,198681804877,748162728797,2823879525331,10681527145369
%N Number of oddlength branches starting at the root in all ordered trees with n edges.
%C Also number of evenlength branches starting at the root in all ordered trees with n+1 edges.  _Emeric Deutsch_, Mar 03 2007
%C Also number of Dyck paths of semilength n+1 with first descent and last ascent of equal size.
%H G. C. Greubel, <a href="/A127540/b127540.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%F a(n) = Sum_{k=0..n} k*A127538(n,k).
%F G.f.: x*C/(1x^2x*Cx^2*C), where C = (1sqrt(14*x))/(2*x) is the Catalan function.
%F a(n) ~ 3*4^(n+1)/(5*sqrt(Pi)*n^(3/2)).  _Vaclav Kotesovec_, Mar 21 2014
%e a(2)=2 because the tree /\ has two oddlength branches starting from the root and the pathtree of length 2 has none.
%e a(2)=2 because the Dyck paths of semilength 3 with first descent and last ascent of same size are UUDUDD and UDUDUD.
%p C:=(1sqrt(14*z))/2/z: g:=z*C/(1z^2z*Cz^2*C): gser:=series(g,z=0,32): seq(coeff(gser,z,n),n=0..29);
%t CoefficientList[Series[x (1  (1  4*x)^(1/2))/(2*x)/(1  x^2  x *(1  (1  4*x)^(1/2)) /(2*x)  x^2*(1  (1  4*x)^(1/2))/(2*x)), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o (PARI) concat([0], Vec(x*(1  (1  4*x)^(1/2))/(2*x)/(1  x^2  x*(1  (1  4*x)^(1/2)) /(2*x)  x^2*(1  (1  4*x)^(1/2))/(2*x)) + O(x^50))) \\ _G. C. Greubel_, Jan 31 2017
%Y Cf. A127538.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Mar 01 2007
