This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127539 #4 Jul 26 2022 11:10:05 %S A127539 1,0,1,0,3,3,16,37,134,411,1411,4747,16500,57671,204380,730032, %T A127539 2629637,9535268,34787215,127585608,470162614,1739952061,6463845941, %U A127539 24096378885,90112499714,337965831635,1270901550454,4790836498608,18100497143361 %N A127539 Number of ordered trees with n edges having no odd-length branches starting at the root. %C A127539 a(n)=A127538(n,0). %F A127539 G.f.=(1+z)*C/(C+z), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function. %F A127539 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 A127539 a(3)=0 because all five ordered trees with 3 edges have at least one odd-length branch starting at the root. %p A127539 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 A127539 Cf. A127538, A000958. %K A127539 nonn %O A127539 0,5 %A A127539 _Emeric Deutsch_, Mar 01 2007