A394117 G.f. A(x) satisfies A(x)^5+(2x-2)*A(x)^4+A(x)^3+(4x^2-2x)*A(x)^2+x^3=0.

0, 1, 2, 7, 30, 144, 744, 4049, 22906, 133537, 797404, 4855941, 30057724, 188635684, 1197865288, 7684428374, 49735550690, 324418503536, 2130770610246, 14080910729781, 93563778237028, 624781101890740, 4190678175148708, 28222633284905036, 190770359607743460

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 only be followed by green vertices, blue vertices can be followed by red or blue 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 A394117(n):
    A = [[0, 0, 1], [1, 1, 0], [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. A394116, A394118.

Sequence in context: A368932 A369160 A243632 * A394129 A196148 A390895

Adjacent sequences: A394114 A394115 A394116 * A394118 A394119 A394120

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved