cp's OEIS Frontend

A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

%I A317829 #30 Aug 31 2020 19:49:40
%S A317829 1,1,4,52,2776,695541,927908528,7303437156115,371421772559819369,
%T A317829 132348505150329265211927,355539706668772869353964510735,
%U A317829 7698296698535929906799439134946965681,1428662247641961794158621629098030994429958386,2405509035205023556420199819453960482395657232596725626
%N A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.
%C A317829 Number of factorizations of the superprimorial A006939(n) into factors > 1. - _Gus Wiseman_, Aug 21 2020
%H A317829 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A317829 a(n) = A317826(A033312(n+1)) = A317826((n+1)!-1) = A001055(A076954(n)).
%F A317829 a(n) = A001055(A006939(n)). - _Gus Wiseman_, Aug 21 2020
%F A317829 a(n) = A318284(A002110(n)). - _Andrew Howroyd_, Aug 31 2020
%e A317829 For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.
%p A317829 g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
%p A317829      `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
%p A317829         g(n/d, d)), d=divisors(n) minus {1, n}))
%p A317829     end:
%p A317829 a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
%p A317829 seq(a(n), n=0..5);  # _Alois P. Heinz_, Jul 26 2020
%t A317829 chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t A317829 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A317829 Table[Length[facs[chern[n]]],{n,3}] (* _Gus Wiseman_, Aug 21 2020 *)
%o A317829 (PARI) \\ See A318284 for count.
%o A317829 a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ _Andrew Howroyd_, Aug 31 2020
%Y A317829 Cf. A033312, A317826.
%Y A317829 Subsequence of A317828.
%Y A317829 A000142 counts submultisets of the same multiset.
%Y A317829 A022915 counts permutations of the same multiset.
%Y A317829 A337069 is the strict case.
%Y A317829 A001055 counts factorizations.
%Y A317829 A006939 lists superprimorials or Chernoff numbers.
%Y A317829 A076716 counts factorizations of factorials.
%Y A317829 A076954 can be used instead of A006939 (cf. A307895, A325337).
%Y A317829 A181818 lists products of superprimorials, with complement A336426.
%Y A317829 Cf. A000178, A002110, A022559, A027423, A124010, A303279, A318284, A322583, A336417, A336496.
%K A317829 nonn
%O A317829 0,3
%A A317829 _Antti Karttunen_, Aug 10 2018
%E A317829 a(0)=1 prepended and a(7) added by _Alois P. Heinz_, Jul 26 2020
%E A317829 a(8)-a(13) from _Andrew Howroyd_, Aug 31 2020