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%I #4 Jul 26 2022 11:10:05
%S 1,0,1,0,3,3,16,37,134,411,1411,4747,16500,57671,204380,730032,
%T 2629637,9535268,34787215,127585608,470162614,1739952061,6463845941,
%U 24096378885,90112499714,337965831635,1270901550454,4790836498608,18100497143361
%N Number of ordered trees with n edges having no odd-length branches starting at the root.
%C a(n)=A127538(n,0).
%F G.f.=(1+z)*C/(C+z), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.
%F D-finite with recurrence (-n+1)*a(n) +2*(n-3)*a(n-1) +(7*n-25)*a(n-2) +(3*n-17)*a(n-3) +(3*n-7)*a(n-4) +2*(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Jul 26 2022
%e a(3)=0 because all five ordered trees with 3 edges have at least one odd-length branch starting at the root.
%p C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)*C/(C+z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..31);
%Y Cf. A127538, A000958.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Mar 01 2007
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