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

0,2

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.

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A157547 A160879 A157562 * A274314 A157592 A213988

Adjacent sequences: A324149 A324150 A324151 * A324153 A324154 A324155

Keyword

nonn

Author

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

Status

approved