A394135 - OEIS

%I #6 Mar 14 2026 15:03:00

%S 0,1,3,16,103,732,5534,43654,355219,2959796,25127182,216566228,

%T 1889981702,16667677502,148309034228,1329840823598,12004376548231,

%U 109001991366916,994928157018222,9123606773971816,84014017887712710,776551886860725700,7202290010661477580

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

%C 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 colors, blue vertices can be followed by red or blue vertices, and green vertices can be followed by red or green vertices.

%H Nathan Fox, <a href="/A394135/b394135.txt">Table of n, a(n) for n = 0..300</a>

%H S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, <a href="https://arxiv.org/abs/2602.16055">Counting Colored Trees</a>, arXiv:2602.16055 [math.CO], 2026.

%o (Python)

%o def A394135(n):

%o     A = [[1, 1, 1], [1, 1, 0], [1, 0, 1]]

%o     if n == 0:

%o         return 0

%o     m = len(A)

%o     output = [[1] for i in range(m)]

%o     for l in range(2, n + 1):

%o         for i in range(m):

%o             term = 0

%o             for k in range(1, l):

%o                 for j in range(m):

%o                     term += A[i][j] * output[i][k - 1] * output[j][l - k - 1]

%o             output[i].append(term)

%o     return output[0][n - 1]

%K nonn

%O 0,3

%A _Nathan Fox_, Mar 11 2026