A324151 a(n) = (2/((n+1)*(n+2)))*multinomial(3*n;n,n,n).

1, 2, 15, 168, 2310, 36036, 612612, 11085360, 210344706, 4143153300, 84106011990, 1750346095680, 37194854533200, 804553314979680, 17671438882589400, 393345439598342880, 8858467087621013610, 201578121034100464500, 4629577513083174001350, 107211268724031397926000

Offset: 0

Author

Michael De Vlieger and N. J. A. Sloane, Mar 01 2019

Keywords

• nonn

Comments

a(n) is an integer, because as Fredes and Sepulveda show, it gives the number of spanning tree decorated quadrangulations rooted in the tree.

For a direct proof, a(n) may also be written as (binomial(3*n,n)/(2*n+1))*(binomial(2*n+2,n)/(n+1)) = A001764(n)*A000108(n+1), and so is an integer. - N. J. A. Sloane, Mar 01 2019

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..704
• Luis Fredes and Avelio Sepulveda, Tree-decorated planar maps, arXiv:1901.04981 [math.CO], 2019. See Remark 4.6.

Crossrefs

Cf. A000108, A001764, A324152.

Programs

Maple

a:= n-> (2/((n+1)*(n+2)))*combinat[multinomial](3*n, n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 25 2022

Mathematica

c[m_, n_] := m Product[1/(n + i), {i, m}] (Multinomial @@ ConstantArray[n, m + 1]); Array[c[2, #] &, 20, 0] (* Michael De Vlieger, Mar 01 2019 *)

Python

from sympy.ntheory import multinomial_coefficients
def A324151(n): return 2*multinomial_coefficients(3,3*n)[(n,n,n)]//(n+1)//(n+2) # Chai Wah Wu, Jan 25 2022