A394161 G.f. A(x) satisfies A(x)^5-3x*A(x)^3+2x^2*A(x)^2+x^2*A(x)-x^3=0.

0, 1, 1, 4, 22, 139, 951, 6855, 51272, 394204, 3096234, 24736726, 200395374, 1642308158, 13591395172, 113424043732, 953433146448, 8065360983795, 68608905310365, 586529022954170, 5036433199693944, 43419795094441061, 375679865662904821, 3261148423057895137

Offset

0,4

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with green 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 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 A394161(n):
    A = [[1, 1, 1], [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[2][n - 1]

Crossrefs

Cf. A394159, A394160.

Sequence in context: A188686 A025756 A366119 * A200731 A193116 A187254

Adjacent sequences: A394158 A394159 A394160 * A394162 A394163 A394164

Keyword

nonn

Author

Nathan Fox, Mar 12 2026

Status

approved