A324152 a(0)=1; for n>0, a(n) = (3/((n+1)*(n+2)*(n+3))) * multinomial(4*n;n,n,n,n).

1, 3, 126, 9240, 900900, 104756652, 13742520792, 1968826448160, 301700280152700, 48756255150603000, 8226155369009738160, 1438285479229504301760, 259131100507849025033760, 47897087290614993606462000, 9050997011303368719799740000

Offset: 0

Author

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

Keywords

• nonn

Comments

It is conjectured that a(n) is always an integer.

If all terms except the first are doubled, we get A324478, which IS known to be integral.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..360
• Luis Fredes, Avelio Sepulveda, Tree-decorated planar maps, arXiv:1901.04981 [math.CO], 2019. See Remark 4.6.

Crossrefs

Cf. A000108, A324151, A324465 (exponent of 2), A324467, A324478.

Programs

Magma

[1] cat [n le 1 select 3 else Self(n-1)*4*(4*n-3)*(4*n-2)*(4*n-1)/((n)^2*(n+3)): n in [1..20]]; // Vincenzo Librandi, Mar 11 2019

Mathematica

c[m_, n_] := m Product[1/(n + i), {i, m}] (Multinomial @@ ConstantArray[n, m + 1]); {1}~Join~Array[c[3, #] &, 14] (* Michael De Vlieger, Mar 01 2019 *)
Flatten[{1, Table[3*(4*n)! / ((n!)^3 * (n+3)!), {n, 1, 15}]}] (* Vaclav Kotesovec, Jul 21 2019 *)

Formula

a(n+1) = a(n)*4*(4*n+1)*(4*n+2)*(4*n+3)/((n+1)^2*(n+4)) for n>0.

From Vaclav Kotesovec, Jul 21 2019: (Start)

For n>0, a(n) = 3*(4*n)! / ((n!)^3 * (n+3)!).

a(n) ~ 3 * 2^(8*n - 1/2) / (Pi^(3/2) * n^(9/2)). (End)