A394139 - OEIS

%I #6 Mar 14 2026 15:03:18

%S 0,1,1,2,7,34,194,1208,7947,54318,381918,2744560,20068358,148829716,

%T 1116773044,8463323632,64683640051,497999144726,3858699223862,

%U 30067745845088,235468907374386,1852285897565836,14629457346782604,115964861319175536,922269694511198782

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

%C 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 blue vertices, blue vertices can be followed by vertices of any color, and green vertices can only be followed by red vertices.

%C Appears that sequence A244062 gives the coefficients on -1/A(x), where A(x) is the generating function for this sequence.

%H Nathan Fox, <a href="/A394139/b394139.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 A394139(n):

%o     A = [[0, 1, 0], [1, 1, 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[2][n - 1]

%Y Cf. A244062, A394138.

%K nonn

%O 0,4

%A _Nathan Fox_, Mar 11 2026