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%I #24 Jan 25 2022 17:14:14
%S 1,2,15,168,2310,36036,612612,11085360,210344706,4143153300,
%T 84106011990,1750346095680,37194854533200,804553314979680,
%U 17671438882589400,393345439598342880,8858467087621013610,201578121034100464500,4629577513083174001350,107211268724031397926000
%N a(n) = (2/((n+1)*(n+2)))*multinomial(3*n;n,n,n).
%C a(n) is an integer, because as Fredes and Sepulveda show, it gives the number of spanning tree decorated quadrangulations rooted in the tree.
%C 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
%H Michael De Vlieger, <a href="/A324151/b324151.txt">Table of n, a(n) for n = 0..704</a>
%H Luis Fredes and Avelio Sepulveda, <a href="https://arxiv.org/abs/1901.04981">Tree-decorated planar maps</a>, arXiv:1901.04981 [math.CO], 2019. See Remark 4.6.
%p a:= n-> (2/((n+1)*(n+2)))*combinat[multinomial](3*n, n$3):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jan 25 2022
%t 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 *)
%o (Python)
%o from sympy.ntheory import multinomial_coefficients
%o def A324151(n): return 2*multinomial_coefficients(3,3*n)[(n,n,n)]//(n+1)//(n+2) # _Chai Wah Wu_, Jan 25 2022
%Y Cf. A000108, A001764, A324152.
%K nonn
%O 0,2
%A _Michael De Vlieger_ and _N. J. A. Sloane_, Mar 01 2019
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