A055335 Number of asymmetric (identity) trees with n nodes and 4 leaves.

1, 3, 8, 14, 25, 40, 62, 89, 127, 173, 233, 304, 393, 497, 624, 769, 942, 1139, 1369, 1627, 1925, 2257, 2635, 3053, 3524, 4042, 4621, 5253, 5954, 6717, 7557, 8466, 9462, 10536, 11706, 12963, 14326, 15786, 17363, 19046, 20857, 22786, 24854

Offset: 9

Author

Christian G. Bower, May 12 2000

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 9..1000
• Index entries for sequences related to trees
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Crossrefs

Column 4 of A055334.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x^9*(1+x+2*x^2-x^3)/((1-x)*(&*[1-x^j: j in [1..4]])) )); // G. C. Greubel, Nov 10 2023

Mathematica

Drop[CoefficientList[Series[x^9*(1+x+2*x^2-x^3)/((1-x)*Product[1-x^j, {j,4}]), {x,0,50}], x], 9] (* G. C. Greubel, Nov 10 2023 *)

SageMath

def A055335_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^9*(1+x+2*x^2-x^3)/((1-x)*product(1-x^j for j in range(1,5))) ).list()
a=A055335_list(50); a[9:] # G. C. Greubel, Nov 10 2023

Formula

G.f.: x^9*(1+x+2*x^2-x^3)/((1-x)^2*(1-x^2)^2*(1+x^2)*(1-x^3)).

a(n) = (-225 -762*n +516*n^2 -100*n^3 +6*n^4)/1152 -(3/128)*(-1)^n*(2*n -11) -(1/16)*(2 -(-1)^n)*(-1)^binomial(n,2) -(1/9)*ChebyshevU(n-1, -1/2) + [n=1]. - G. C. Greubel, Nov 10 2023