A394158 G.f. A(x) satisfies A(x)^7-4*A(x)^6+(2x+6)*A(x)^5-(7x+4)*A(x)^4+(3x^2+8x+1)*A(x)^3-(5x^2+3x)*A(x)^2+(x^3+3x^2)*A(x)-x^3=0.

0, 1, 2, 8, 41, 241, 1549, 10609, 76205, 568016, 4360514, 34285365, 274948040, 2241563703, 18530933242, 155024266777, 1310202864014, 11171856536985, 96000675068603, 830578796200132, 7229453998159332, 63264663039028941, 556292931608639795, 4912732295552118304

Offset

0,3

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with green root where red vertices can be followed by vertices of any colors, blue vertices can be followed red or green 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 A394158(n):
    A = [[1, 1, 1], [1, 0, 1], [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[2][n - 1]

Crossrefs

Cf. A394156, A394157.

Sequence in context: A381817 A294084 A381826 * A394150 A177340 A067119

Adjacent sequences: A394155 A394156 A394157 * A394159 A394160 A394161

Keyword

nonn

Author

Nathan Fox, Mar 12 2026

Status

approved