A394126 G.f. A(x) satisfies A(x)^6-3*A(x)^5+(x+3)*A(x)^4-(4x+1)*A(x)^3+4x*A(x)^2-x*A(x)+x^2=0.

0, 1, 3, 14, 77, 464, 2968, 19806, 136407, 962618, 6925550, 50611024, 374661606, 2803693956, 21174556020, 161187646856, 1235473430325, 9526917196888, 73855947802228, 575279950326078, 4500074754843640, 35336594070610722, 278443821633875298, 2201014965056842878

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 vertices of any color, and green vertices can only be followed by 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 A394126(n):
    A = [[0, 1, 0], [1, 1, 1], [0, 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[1][n - 1]

Crossrefs

Cf. A394125.

Sequence in context: A394120 A048779 A369477 * A394138 A393906 A052186

Adjacent sequences: A394123 A394124 A394125 * A394127 A394128 A394129

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved