A394157 G.f. A(x) satisfies A(x)^7+x*A(x)^6-2x*A(x)^5-x^2*A(x)^4+2x^2*A(x)^3-x^3*A(x)+x^4=0.

0, 1, 2, 9, 52, 340, 2395, 17736, 136156, 1074029, 8654041, 70930051, 589550373, 4957792102, 42107307872, 360675852281, 3112264077266, 27029371452916, 236083608533439, 2072482749835020, 18275920358494505, 161820500888016391, 1438085180352500918

Offset

0,3

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with blue root where red vertices can be followed by vertices of any colors, blue vertices can be followed red or green vertices, and green vertices can be followed by blue or green 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 A394157(n):
    A = [[1, 1, 1], [1, 0, 1], [0, 1, 1]]
    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[1][n - 1]

Crossrefs

Cf. A394156, A394158.

Sequence in context: A301928 A394146 A069271 * A381782 A305987 A355789

Adjacent sequences: A394154 A394155 A394156 * A394158 A394159 A394160

Keyword

nonn

Author

Nathan Fox, Mar 12 2026

Status

approved