A394111 - OEIS

%I #14 Mar 14 2026 15:00:37

%S 3,4,13,54,253,1274,6732,36822,206701,1184040,6893445,40670760,

%T 242632908,1461177064,8870797064,54233183862,333605097741,

%U 2063260443860,12822404577255,80031319222470,501463396164885,3153170215068930,19890529451148720,125839151727230760

%N a(n) = binomial(3*n, n)*(5*n+3)/((n+1)*(2*n+1)).

%C a(n-1) equals the number of n-vertex planar rooted trees with vertices colored red, blue, and green where red and green vertices can only be followed by blue vertices, and blue vertices can be followed by red or blue vertices.

%H Nathan Fox, <a href="/A394111/b394111.txt">Table of n, a(n) for n = 0..1000</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) = A006013(n) + 2*A001764(n).

%F G.f.: 2*(1 - hypergeom([-2/3, -1/3], [1/2], 27*x) + 5*sqrt(3*x)*sin(arcsin(3*sqrt(3*x)/2)/3))/(3*x). - _Stefano Spezia_, Mar 12 2026

%o (Python)

%o from math import comb

%o def A394111(n):

%o     return comb(3*n, n)*(5*n+3)//(n+1)//(2*n+1)

%Y Cf. A001764, A006013, A394110.

%K nonn,easy

%O 0,1

%A _Nathan Fox_, Mar 11 2026