A394111 a(n) = binomial(3*n, n)*(5*n+3)/((n+1)*(2*n+1)).

3, 4, 13, 54, 253, 1274, 6732, 36822, 206701, 1184040, 6893445, 40670760, 242632908, 1461177064, 8870797064, 54233183862, 333605097741, 2063260443860, 12822404577255, 80031319222470, 501463396164885, 3153170215068930, 19890529451148720, 125839151727230760

Offset

0,1

Comments

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.

Links

Nathan Fox, Table of n, a(n) for n = 0..1000

S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, Counting Colored Trees, arXiv:2602.16055 [math.CO], 2026.

Formula

a(n) = A006013(n) + 2*A001764(n).

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

Prog

(Python)
from math import comb
def A394111(n):
    return comb(3*n, n)*(5*n+3)//(n+1)//(2*n+1)

Crossrefs

Cf. A001764, A006013, A394110.

Sequence in context: A243764 A201821 A001056 * A122151 A294384 A216868

Adjacent sequences: A394108 A394109 A394110 * A394112 A394113 A394114

Keyword

nonn,easy

Author

Nathan Fox, Mar 11 2026

Status

approved