A229414 Number of set partitions of {1,...,3n} into sets of size at most 3.

1, 5, 166, 12644, 1680592, 341185496, 97620050080, 37286121988256, 18280749571449664, 11168256342434121152, 8306264068494786829696, 7380771881944947770497280, 7715405978050522488223499776, 9365880670184268387214967727104, 13058232187415887547449498864463872

Offset: 0

Author

Alois P. Heinz, Sep 22 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Row n=3 of A229243.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [1, 5, 166][n+1],
      ((108*n^2-72*n+4)*a(n-1)-6*(n-1)*(3*n-5)*(27*n^2-48*n+10)*a(n-2)
       +9*(n-1)*(n-2)*(3*n-1)*(3*n-7)*(3*n-5)*(3*n-8)*a(n-3))/8)
    end:
seq(a(n), n=0..20);

Mathematica

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n - k, k]; For[j = k - 1, j >= 1, j--, g = g(n-j)/j + G[n - j, k]]; g]];
a[n_] := G[3n, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz in A229243 *)

Formula

a(n) = (3n)! * [x^(3n)] exp(x + x^2/2 + x^3/6).

a(n) = A001680(3n) = A229223(3n,3).