A394141 G.f. A(x) satisfies A(x)^5-2*A(x)^4+(2x+1)*A(x)^3-4x*A(x)^2+x*A(x)-x^2=0.

0, 1, 3, 15, 89, 577, 3954, 28159, 206344, 1545851, 11787445, 91191974, 714062041, 5648777429, 45079529911, 362495811596, 2934348587956, 23892641086459, 195557929860835, 1608071398391111, 13278516834574055, 110060894070407408, 915386559028780435

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 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 A394141(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[1][n - 1]

Crossrefs

Cf. A394140, A394142.

Sequence in context: A287511 A231257 A074541 * A394153 A368974 A394131

Adjacent sequences: A394138 A394139 A394140 * A394142 A394143 A394144

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved