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A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

%I A320291 #9 Oct 26 2018 00:52:42
%S A320291 1,0,0,0,1,1,3,3,7,8,15,19,36,46,79,110,181,254,407,580,907,1309,2004,
%T A320291 2909,4410,6407,9599,13984,20782,30252,44677,64967,95414,138563,
%U A320291 202527,293583,427442,618337,897023,1295020,1872696,2697777,3889964,5591917,8041593,11535890
%N A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.
%H A320291 Andrew Howroyd, <a href="/A320291/b320291.txt">Table of n, a(n) for n = 0..1000</a>
%F A320291 Euler transform of A083751. - _Andrew Howroyd_, Oct 25 2018
%e A320291 The a(4) = 1 through a(10) = 15 multiset partitions:
%e A320291   ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
%e A320291                   ((33))   ((34))   ((35))      ((36))      ((37))
%e A320291                   ((222))  ((223))  ((44))      ((45))      ((46))
%e A320291                                     ((224))     ((225))     ((55))
%e A320291                                     ((233))     ((234))     ((226))
%e A320291                                     ((2222))    ((333))     ((235))
%e A320291                                     ((22)(22))  ((2223))    ((244))
%e A320291                                                 ((22)(23))  ((334))
%e A320291                                                             ((2224))
%e A320291                                                             ((2233))
%e A320291                                                             ((22222))
%e A320291                                                             ((22)(24))
%e A320291                                                             ((22)(33))
%e A320291                                                             ((23)(23))
%e A320291                                                             ((22)(222))
%t A320291 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320291 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320291 Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]
%o A320291 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320291 seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ _Andrew Howroyd_, Oct 25 2018
%Y A320291 Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.
%K A320291 nonn
%O A320291 0,7
%A A320291 _Gus Wiseman_, Oct 09 2018
%E A320291 Terms a(21) and beyond from _Andrew Howroyd_, Oct 25 2018