A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.

1, 3, 7, 21, 71, 253, 925, 3433, 12871, 48621, 184757, 705433, 2704157, 10400601, 40116601, 155117521, 601080391, 2333606221, 9075135301, 35345263801, 137846528821, 538257874441, 2104098963721, 8233430727601, 32247603683101, 126410606437753, 495918532948105

Offset: 0

Author

Alois P. Heinz, Jun 21 2014

Keywords

• nonn

Examples

			a(2) = 7: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A007318, A065567, A242451, A245732.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<3, 2^(n+1)-1, ((15*n^2-31*n+12) *a(n-1)
       -2*(3*n-2)*(2*n-3) *a(n-2)) / ((3*n-5)*n))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := a[n] = If[n < 3, 2^(n+1) - 1, ((15*n^2 - 31*n + 12)*a[n-1] - 2*(3*n - 2)*(2*n - 3)*a[n-2])/((3*n - 5)*n)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2014, after Alois P. Heinz *)

Sage

A244174 = lambda m: SetPartitions(2*m,[2*m]).cardinality()+2*SetPartitions(2*m,[m,m]).cardinality()
[1] + [A244174(m) for m in (1..26)] # Peter Luschny, Aug 02 2015

Formula

a(n) = A242451(3n,n).

Recurrence: see Maple program.

For n>0, a(n) = 1 + C(2n,n) = 1 + A000984(n). - Vaclav Kotesovec, Jun 21 2014

G.f.: 1/(sqrt(1-4*x)) + x/(1-x). - Alois P. Heinz, Jun 22 2014

a(n) = A245732(2n,n). - Alois P. Heinz, Jul 30 2014

a(n) = A065567(2n,n) for n>=1. - Alois P. Heinz, Sep 05 2023