A394110 a(n) = binomial(3*n, n)*(7*n+3)/((n+1)*(2*n+1)).

3, 5, 17, 72, 341, 1729, 9180, 50388, 283613, 1628055, 9494745, 56097600, 335064492, 2019862412, 12273294568, 75092100732, 462223930605, 2860429251715, 17785916026515, 111063871574040, 696206462636685, 4379403076484625, 27635514370180080

Offset

0,1

Comments

a(n-1) equals the of n-vertex planar rooted trees with vertices colored red, blue, and green where red and blue vertices can be followed by red or green vertices, and green vertices can only be followed by red 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) = 2*A006013(n) + A001764(n).

G.f.: 2*(2 - 2*hypergeom([-2/3, -1/3], [1/2], 27*x]) + 7*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 A394110(n):
    return comb(3*n, n)*(7*n+3)//(n+1)//(2*n+1)

Crossrefs

Cf. A001764, A006013, A394111.

Sequence in context: A319908 A356256 A232238 * A102295 A365518 A227335

Adjacent sequences: A394107 A394108 A394109 * A394111 A394112 A394113

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Status

approved