A394140 G.f. A(x) satisfies (x+1)*A(x)^5-x*A(x)^4-(2x^2+x)*A(x)^3+(2x^3+3x^2)*A(x)^2-3x^3*A(x)+x^4=0.

0, 1, 1, 4, 22, 138, 930, 6561, 47800, 356728, 2712758, 20944303, 163738947, 1293610671, 10312170488, 82843804507, 670045658032, 5451693184584, 44591169828366, 366447128131373, 3024200357007623, 25053501077514721, 208272200149758543, 1736864093227610857

Offset

0,4

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with red root where red vertices can only be followed by blue vertices, blue vertices can be followed by vertices of any colors, and green vertices can be followed by red 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 A394140(n):
    A = [[0, 1, 0], [1, 1, 1], [1, 0, 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[0][n - 1]

Crossrefs

Cf. A394141, A394142.

Sequence in context: A091638 A142984 A283055 * A097593 A386553 A188686

Adjacent sequences: A394137 A394138 A394139 * A394141 A394142 A394143

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved