A143360 Sum of root degrees of all symmetric ordered trees with n edges.

1, 3, 5, 12, 20, 45, 77, 168, 294, 630, 1122, 2376, 4290, 9009, 16445, 34320, 63206, 131274, 243542, 503880, 940576, 1939938, 3640210, 7488432, 14115100, 28973100, 54826020, 112326480, 213286590, 436268025, 830905245, 1697168160, 3241119750, 6611884290

Offset: 1

Author

Emeric Deutsch, Aug 15 2008

Keywords

• nonn

Crossrefs

Cf. A000108, A129869 and A143359.

Programs

Maple

C := z -> (1/2-(1/2)*sqrt(1-4*z))/z: G := z*C(z^2)^2*(1+2*z*C(z^2))/(1-z*C(z^2)): Gser := series(G, z=0, 40): seq(coeff(Gser, z, n), n=1..34);

Mathematica

Module[{nmax = 33, G, C}, G = z*C[z^2]^2*(1 + 2*z*C[z^2])/(1 - z*C[z^2]); C[z_] = (1/2-(1/2)*Sqrt[1-4*z])/z; CoefficientList[G/z + O[z]^nmax, z]] (* Jean-François Alcover, Apr 09 2024 *)

Formula

G.f.: z*C(z^2)^2*(1+2*z*C(z^2))/(1-z*C(z^2)), where C(z)=(1-sqrt(1-4*z))/(2*z) is the g.f. of the Catalan numbers (A000108).

a(n) = Sum_{k=1..n} k * A143359(n,k).

D-finite with recurrence 2*(n+3)*a(n) +(-n-5)*a(n-1) +(-11*n-3)*a(n-2) +2*(2*n+1)*a(n-3) +12*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 24 2022