A320291 - OEIS

%I #9 Oct 26 2018 00:52:42

%S 1,0,0,0,1,1,3,3,7,8,15,19,36,46,79,110,181,254,407,580,907,1309,2004,

%T 2909,4410,6407,9599,13984,20782,30252,44677,64967,95414,138563,

%U 202527,293583,427442,618337,897023,1295020,1872696,2697777,3889964,5591917,8041593,11535890

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

%H Andrew Howroyd, <a href="/A320291/b320291.txt">Table of n, a(n) for n = 0..1000</a>

%F Euler transform of A083751. - _Andrew Howroyd_, Oct 25 2018

%e The a(4) = 1 through a(10) = 15 multiset partitions:

%e   ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))

%e                   ((33))   ((34))   ((35))      ((36))      ((37))

%e                   ((222))  ((223))  ((44))      ((45))      ((46))

%e                                     ((224))     ((225))     ((55))

%e                                     ((233))     ((234))     ((226))

%e                                     ((2222))    ((333))     ((235))

%e                                     ((22)(22))  ((2223))    ((244))

%e                                                 ((22)(23))  ((334))

%e                                                             ((2224))

%e                                                             ((2233))

%e                                                             ((22222))

%e                                                             ((22)(24))

%e                                                             ((22)(33))

%e                                                             ((23)(23))

%e                                                             ((22)(222))

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ _Andrew Howroyd_, Oct 25 2018

%Y Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

%K nonn

%O 0,7

%A _Gus Wiseman_, Oct 09 2018

%E Terms a(21) and beyond from _Andrew Howroyd_, Oct 25 2018