A394137 - OEIS

%I #8 Jun 04 2026 04:02:12

%S 0,1,2,6,21,81,334,1444,6463,29695,139255,663817,3207057,15668163,

%T 77274843,384221651,1923893753,9692937761,49101121091,249935367839,

%U 1277744991383,6557730926365,33775001481244,174514216113442,904355547895403,4699102486549625

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

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

%H Nathan Fox, <a href="/A394137/b394137.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.

%F a(n) ~ 11^(1/4) * 2^(n - 5/2) * 3^(2*n - 7/4) / (sqrt(Pi) * n^(3/2) * (9 - sqrt(33))^(n - 3/2)). - _Vaclav Kotesovec_, Jun 04 2026

%o (Python)

%o def A394137(n):

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

%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[1][n - 1]

%Y Cf. A394136.

%K nonn

%O 0,3

%A _Nathan Fox_, Mar 11 2026