A394119 G.f. A(x) satisfies A(x)^4+(2x-1)*A(x)^3+(x^2+2x)*A(x)^2+(2x^2-x)*A(x)+x^2=0.

0, 1, 3, 13, 66, 366, 2150, 13159, 83060, 536990, 3538667, 23684774, 160580717, 1100573304, 7612835955, 53078563117, 372638335430, 2631988112768, 18689773748375, 133350103375528, 955513605765357, 6873080290918896, 49611190313180829, 359240625644681554

Offset

0,3

Comments

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

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 A394119(n):
    A = [[1, 1, 1], [0, 1, 0], [0, 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[0][n - 1]

Crossrefs

Sequence in context: A394145 A112807 A219537 * A045743 A381875 A110530

Adjacent sequences: A394116 A394117 A394118 * A394120 A394121 A394122

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved