A394127 G.f. A(x) satisfies A(x)^3+(x-2)*A(x)^2+(1-x)*A(x)+x^2-x=0.

0, 1, 2, 8, 38, 198, 1096, 6330, 37722, 230248, 1432230, 9046382, 57865712, 374084722, 2440256850, 16042605800, 106181903430, 706975917526, 4731960602456, 31820872887146, 214885891901898, 1456631639356008, 9907950315975622, 67604709705143518, 462607860107453920

Offset

0,3

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with red root where red vertices can be followed by red or blue vertices, blue vertices can be followed by red or green vertices, and green vertices can only occur as leaves.

Links

Nathan Fox, Table of n, a(n) for n = 0..300

S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, Counting Colored Trees, arXiv:2602.16055 [math.CO], 2026.

Formula

a(n) ~ (3 + 7/(4*cos(Pi/7)-3))^(n - 1/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2026

Prog

(Python)
def A394127(n):
    A = [[1, 1, 0], [1, 0, 1], [0, 0, 0]]
    if n == 0:
        return 0
    m = len(A)
    output = [[1] for i in range(m)]
    for l in range(2, n + 1):
        for i in range(m):
            term = 0
            for k in range(1, l):
                for j in range(m):
                    term += A[i][j] * output[i][k - 1] * output[j][l - k - 1]
            output[i].append(term)
    return output[0][n - 1]

Crossrefs

Sequence in context: A271934 A364723 A372107 * A266797 A369208 A234939

Adjacent sequences: A394124 A394125 A394126 * A394128 A394129 A394130

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved