A394121 G.f. A(x) satisfies A(x)^8-A(x)^7+7x*A(x)^6-4x*A(x)^5+13x^2*A(x)^4-4x^2*A(x)^3+7x^3*A(x)^2-x^3*A(x)+x^4=0.

0, 1, 3, 15, 87, 544, 3566, 24165, 167904, 1189887, 8568923, 62539975, 461652474, 3441166512, 25868639637, 195913687174, 1493490598805, 11451708404208, 88267258280478, 683535077436051, 5315642601664337, 41496627862478209, 325073931504325482, 2554656623330956878

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

The integer coefficients on the even powers in the functional equation defining this sequence equal the odd-indexed terms from A147987(11) through A147987(19).

The integer coefficients on the odd powers in the functional equation defining this sequence equal the negatives of the even-indexed terms from A147988(8) through A147988(14).

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) ~ 29^(2*n - 1/2) / (3 * sqrt(7*Pi) * n^(3/2) * 2^(2*n) * 5^(2*n - 3/2)). - Vaclav Kotesovec, Jun 04 2026

Prog

(Python)
def A394121(n):
    A = [[1, 0, 0], [1, 1, 0], [1, 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. A007852, A147987, A147988.

Sequence in context: A168503 A370184 A089022 * A359797 A246538 A132371

Adjacent sequences: A394118 A394119 A394120 * A394122 A394123 A394124

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved