A271715 Number of set partitions of [3n] with minimal block length multiplicity equal to n.

1, 4, 55, 1540, 67375, 4239235, 383563180, 51925673800, 10652498631775, 3139051466175625, 1228555090548911125, 602267334323068414000, 357161594247065690582500, 250870551734754490461422500, 205672479804595549379158525000, 194557626586812183102927448930000

Offset

0,2

Links

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

Formula

a(n) = A271424(3n,n).

Recursion: see Maple program.

For n>0, a(n) = (3^n + n!)*(3*n)! / (6^n * (n!)^2). - Vaclav Kotesovec, Apr 16 2016

a(n) = A372722(3n,n). - Alois P. Heinz, May 11 2024

Maple

a:= proc(n) option remember; `if`(n<5,
      [1, 4, 55, 1540, 67375][n+1], ((2*(3*n-2))*
       (3*n-1)*(n^2-n-9)*a(n-1) -(3*(n-3))*(3*n-1)*
       (3*n-4)*(3*n-2)*(3*n-5)*a(n-2))/(4*n*(n-4)))
    end:
seq(a(n), n=0..20);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i&, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]];
a[n_] := If[n==0, 1, b[3n, 3n, n] - b[3n, 3n, n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz in A271424 *)

Crossrefs

Cf. A271424, A372722.

Sequence in context: A206384 A380919 A379662 * A099122 A001500 A246968

Adjacent sequences: A271712 A271713 A271714 * A271716 A271717 A271718

Keyword

nonn

Author

Alois P. Heinz, Apr 12 2016

Status

approved