A394115 G.f. A(x) satisfies A(x)^3+(2x-2)*A(x)^2+(1-x)*A(x)+x^2-x=0.

0, 1, 2, 7, 29, 132, 639, 3230, 16855, 90121, 491197, 2719051, 15245058, 86396108, 494101906, 2848033464, 16528607591, 96500097652, 566395507497, 3340116683850, 19780686390699, 117592135034870, 701482912853093, 4197815714462468, 25193001701058114

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

Appears that the inverse binomial transform starting from index n=2 yields sequence A217596.

Appears that the inversion starting from index n=1 yields sequence A006026.

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) ~ 2^(5*n-4) / (3*sqrt(Pi) * n^(3/2) * 5^(n - 3/2)). - Vaclav Kotesovec, Jun 04 2026

Prog

(Python)
def A394115(n):
    A = [[0, 0, 1], [1, 0, 0], [0, 1, 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. A006026, A217596.

Sequence in context: A232971 A366084 A368935 * A110576 A074600 A064641

Adjacent sequences: A394112 A394113 A394114 * A394116 A394117 A394118

Keyword

nonn,changed

Author

Nathan Fox, Mar 11 2026

Status

approved