A394143 G.f. A(x) satisfies A(x)^4-3*A(x)^3+(3-x)*A(x)^2-A(x)-x^2+x=0.

0, 1, 2, 8, 39, 210, 1203, 7192, 44362, 280250, 1804104, 11792186, 78052758, 522114602, 3524075051, 23970980364, 164154325319, 1130807642570, 7830693456708, 54480359509912, 380626675444910, 2669315423325620, 18784031743471065, 132596873090317300

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 blue or green vertices, blue vertices can be followed by red or green vertices, and green vertices can only be followed by red vertices.

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.

Prog

(Python)
def A394143(n):
    A = [[1, 1, 0], [1, 0, 1], [1, 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

Cf. A394144, A394145.

Sequence in context: A394118 A394128 A394134 * A366049 A218321 A236339

Adjacent sequences: A394140 A394141 A394142 * A394144 A394145 A394146

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved