cp's OEIS Frontend

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

%I A330452 #9 Dec 29 2019 16:01:05
%S A330452 1,1,2,7,13,34,81,175,403,890,1977,4262,9356,19963,42573,90865,191206,
%T A330452 401803,837898,1744231,3607504,7436628,15254309,31185686,63552725,
%U A330452 128963236,260933000,526140540,1057927323,2120500885,4239012067,8449746787,16799938614
%N A330452 Number of set partitions of strict multiset partitions of integer partitions of n.
%C A330452 Number of sets of disjoint nonempty sets of nonempty multisets of positive integers with total sum n.
%H A330452 Andrew Howroyd, <a href="/A330452/b330452.txt">Table of n, a(n) for n = 0..500</a>
%F A330452 a(n) = Sum_{0 <= k <= n} A330463(n,k) * A000110(k).
%e A330452 The a(4) = 13 partitions:
%e A330452   ((4))  ((22))  ((31))      ((211))      ((1111))
%e A330452                  ((1)(3))    ((1)(21))    ((1)(111))
%e A330452                  ((1))((3))  ((2)(11))    ((1))((111))
%e A330452                              ((1))((21))
%e A330452                              ((2))((11))
%t A330452 ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t A330452 Table[Length[Select[ppl[n,3],UnsameQ@@Join@@#&]],{n,0,10}]
%o A330452 (PARI) \\ here BellP is A000110 as series.
%o A330452 BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
%o A330452 seq(n)={my(b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ _Andrew Howroyd_, Dec 29 2019
%Y A330452 Cf. A001970, A007713, A050343, A063834, A089259, A261049, A271619, A279375, A294617, A318565, A323787-A323795, A330452-A330459, A330460.
%K A330452 nonn
%O A330452 0,3
%A A330452 _Gus Wiseman_, Dec 16 2019
%E A330452 Terms a(18) and beyond from _Andrew Howroyd_, Dec 29 2019