A127539 Number of ordered trees with n edges having no odd-length branches starting at the root.
1, 0, 1, 0, 3, 3, 16, 37, 134, 411, 1411, 4747, 16500, 57671, 204380, 730032, 2629637, 9535268, 34787215, 127585608, 470162614, 1739952061, 6463845941, 24096378885, 90112499714, 337965831635, 1270901550454, 4790836498608, 18100497143361
Offset: 0
Keywords
Examples
a(3)=0 because all five ordered trees with 3 edges have at least one odd-length branch starting at the root.
Programs
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Maple
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);
Formula
G.f.=(1+z)*C/(C+z), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.
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
Comments