A394139 G.f. A(x) satisfies (x-1)*A(x)^4+(-2x^2+3x-1)*A(x)^3+(-2x^2+3x)*A(x)^2-3x^2*A(x)+x^3=0.

0, 1, 1, 2, 7, 34, 194, 1208, 7947, 54318, 381918, 2744560, 20068358, 148829716, 1116773044, 8463323632, 64683640051, 497999144726, 3858699223862, 30067745845088, 235468907374386, 1852285897565836, 14629457346782604, 115964861319175536, 922269694511198782

Offset

0,4

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with green root where red vertices can only be followed by blue vertices, blue vertices can be followed by vertices of any color, and green vertices can only be followed by red vertices.

Appears that sequence A244062 gives the coefficients on -1/A(x), where A(x) is the generating function for this sequence.

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 A394139(n):
    A = [[0, 1, 0], [1, 1, 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[2][n - 1]

Crossrefs

Cf. A244062, A394138.

Sequence in context: A206240 A289720 A190631 * A326560 A199475 A241599

Adjacent sequences: A394136 A394137 A394138 * A394140 A394141 A394142

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved