cp's OEIS Frontend

A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

%I A330453 #8 Jul 18 2021 20:55:12
%S A330453 1,1,3,9,23,62,161,410,1031,2579,6359,15575,37830,91241,218581,520544,
%T A330453 1232431,2902644,6802178,15866054,36844016,85202436,196251933,
%U A330453 450341874,1029709478,2346409350,5329371142,12066816905,27240224766,61317231288,137643961196
%N A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.
%C A330453 Number of sets of nonempty multisets of nonempty multisets of positive integers with total sum n.
%H A330453 Alois P. Heinz, <a href="/A330453/b330453.txt">Table of n, a(n) for n = 0..3853</a>
%F A330453 Weigh transform of A001970. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.
%e A330453 The a(4) = 23 partitions:
%e A330453   ((4))  ((22))    ((31))      ((211))        ((1111))
%e A330453          ((2)(2))  ((1)(3))    ((1)(21))      ((1)(111))
%e A330453                    ((1))((3))  ((2)(11))      ((11)(11))
%e A330453                                ((1)(1)(2))    ((1))((111))
%e A330453                                ((1))((21))    ((1)(1)(11))
%e A330453                                ((2))((11))    ((1))((1)(11))
%e A330453                                ((1))((1)(2))  ((1)(1)(1)(1))
%e A330453                                ((2))((1)(1))  ((11))((1)(1))
%e A330453                                               ((1))((1)(1)(1))
%p A330453 with(numtheory): with(combinat):
%p A330453 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A330453       numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A330453     end:
%p A330453 a:= proc(n) a(n):= `if`(n<2, 1, add(a(n-k)*add(b(d)
%p A330453       *d*(-1)^(k/d+1), d=divisors(k)), k=1..n)/n)
%p A330453     end:
%p A330453 seq(a(n), n=0..32);  # _Alois P. Heinz_, Jul 18 2021
%t A330453 ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t A330453 Table[Length[Select[ppl[n,3],UnsameQ@@#&]],{n,0,10}]
%Y A330453 The not necessarily strict case is A007713.
%Y A330453 Cf. A001055, A001970, A050336, A050343, A089259, A261049, A271619, A316980, A318566, A323787-A323795, A330452-A330459, A330461, A330463.
%K A330453 nonn
%O A330453 0,3
%A A330453 _Gus Wiseman_, Dec 17 2019