A394137 G.f. A(x) satisfies (1-x)*A(x)^3+(3x-2)*A(x)^2+(1-x)*A(x)+x^2-x=0.

0, 1, 2, 6, 21, 81, 334, 1444, 6463, 29695, 139255, 663817, 3207057, 15668163, 77274843, 384221651, 1923893753, 9692937761, 49101121091, 249935367839, 1277744991383, 6557730926365, 33775001481244, 174514216113442, 904355547895403, 4699102486549625

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 red or green vertices, and green vertices can only be followed by red 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.

Formula

a(n) ~ 11^(1/4) * 2^(n - 5/2) * 3^(2*n - 7/4) / (sqrt(Pi) * n^(3/2) * (9 - sqrt(33))^(n - 3/2)). - Vaclav Kotesovec, Jun 04 2026

Prog

(Python)
def A394137(n):
    A = [[0, 1, 0], [1, 0, 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[1][n - 1]

Crossrefs

Cf. A394136.

Sequence in context: A279565 A150214 A150215 * A328434 A150216 A150217

Adjacent sequences: A394134 A394135 A394136 * A394138 A394139 A394140

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved