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

%I A244174 #25 Sep 05 2023 20:35:57
%S A244174 1,3,7,21,71,253,925,3433,12871,48621,184757,705433,2704157,10400601,
%T A244174 40116601,155117521,601080391,2333606221,9075135301,35345263801,
%U A244174 137846528821,538257874441,2104098963721,8233430727601,32247603683101,126410606437753,495918532948105
%N A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.
%H A244174 Alois P. Heinz, <a href="/A244174/b244174.txt">Table of n, a(n) for n = 0..1000</a>
%F A244174 a(n) = A242451(3n,n).
%F A244174 Recurrence: see Maple program.
%F A244174 For n>0, a(n) = 1 + C(2n,n) = 1 + A000984(n). - _Vaclav Kotesovec_, Jun 21 2014
%F A244174 G.f.: 1/(sqrt(1-4*x)) + x/(1-x). - _Alois P. Heinz_, Jun 22 2014
%F A244174 a(n) = A245732(2n,n). - _Alois P. Heinz_, Jul 30 2014
%F A244174 a(n) = A065567(2n,n) for n>=1. - _Alois P. Heinz_, Sep 05 2023
%e A244174 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].
%p A244174 a:= proc(n) option remember;
%p A244174       `if`(n<3, 2^(n+1)-1, ((15*n^2-31*n+12) *a(n-1)
%p A244174        -2*(3*n-2)*(2*n-3) *a(n-2)) / ((3*n-5)*n))
%p A244174     end:
%p A244174 seq(a(n), n=0..30);
%t A244174 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_ *)
%o A244174 (Sage)
%o A244174 A244174 = lambda m: SetPartitions(2*m,[2*m]).cardinality()+2*SetPartitions(2*m,[m,m]).cardinality()
%o A244174 [1] + [A244174(m) for m in (1..26)] # _Peter Luschny_, Aug 02 2015
%Y A244174 Cf. A000984, A007318, A065567, A242451, A245732.
%K A244174 nonn
%O A244174 0,2
%A A244174 _Alois P. Heinz_, Jun 21 2014