A394133 G.f. A(x) satisfies A(x)^5-2*A(x)^4+(4x+1)*A(x)^3-(3x^2+3x)*A(x)^2+(x^3+3x^2)*A(x)-x^3=0.

0, 1, 1, 3, 12, 56, 288, 1583, 9129, 54578, 335522, 2108754, 13492814, 87615489, 575986999, 3826311988, 25646949717, 173243532072, 1178199786364, 8060658132743, 55439470986851, 383105416597112, 2658636183714792, 18520888690187382, 129471023915349560

Offset

0,4

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 only be followed red vertices, 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 A394133(n):
    A = [[0, 1, 1], [1, 0, 0], [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[1][n - 1]

Crossrefs

Cf. A394132, A394134.

Sequence in context: A369482 A366097 A074533 * A000257 A301418 A224922

Adjacent sequences: A394130 A394131 A394132 * A394134 A394135 A394136

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved