A394144 G.f. A(x) satisfies (1-x)*A(x)^4+x*A(x)^3+(4x^2-2x)*A(x)^2+x^2*A(x)+x^3=0.

0, 1, 2, 7, 31, 157, 864, 5024, 30370, 188941, 1201771, 7779443, 51083531, 339437226, 2278112737, 15420424861, 105152580575, 721670737533, 4981038781815, 34552941731249, 240769486524417, 1684502378190617, 11828416221973412, 83333705758153522, 588880900214492672

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 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 A394144(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[1][n - 1]

Crossrefs

Cf. A394143, A394145.

Sequence in context: A394132 A030873 A030913 * A181066 A325452 A030945

Adjacent sequences: A394141 A394142 A394143 * A394145 A394146 A394147

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved