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A244174
Number of compositions of 3n in which the minimal multiplicity of parts equals n.
9
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
0,2
LINKS
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
EXAMPLE
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].
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 21 2014
STATUS
approved