A394116 G.f. A(x) satisfies A(x)^5-(2x+3)*A(x)^4+(7x+1)*A(x)^3-(4x^2+3x)*A(x)^2+3x^2*A(x)-x^3=0.

0, 1, 1, 3, 13, 66, 364, 2111, 12664, 77853, 487589, 3098863, 19930627, 129458464, 847941004, 5593838322, 37132420061, 247835922698, 1662152295702, 11195577583449, 75700837876997, 513654611900204, 3496381454815240, 23868386315806536, 163372487283786302

Offset

0,4

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with red 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 A394116(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[0][n - 1]

Crossrefs

Cf. A394117, A394118.

Sequence in context: A156181 A260783 A373932 * A228987 A394145 A112807

Adjacent sequences: A394113 A394114 A394115 * A394117 A394118 A394119

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved