A394142 G.f. A(x) satisfies (x+1)*A(x)^5-(x+2)*A(x)^4+(2x^2+3x+1)*A(x)^3-(x^2+3x)*A(x)^2+3x^2*A(x)-x^3=0.

0, 1, 2, 7, 31, 158, 888, 5372, 34430, 231116, 1610639, 11572782, 85262374, 641278125, 4906551925, 38082333652, 299161679020, 2374285935782, 19009255256765, 153349252634092, 1245249010912888, 10170369472358633, 83489241602340722, 688481467240020797

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 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 A394142(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[2][n - 1]

Crossrefs

Cf. A394140, A394141.

Sequence in context: A030945 A088554 A107595 * A193320 A030882 A273957

Adjacent sequences: A394139 A394140 A394141 * A394143 A394144 A394145

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved