Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Aug 22 2019 09:58:43
%S 1,0,1,0,1,1,0,1,0,2,0,1,1,1,1,1,0,1,0,2,0,3,0,1,0,1,1,3,0,1,1,2,1,1,
%T 0,1,0,3,0,3,0,4,0,1,0,3,0,1,1,3,1,3,0,3,2,1,0,4,0,1,1,1,0,1,0,5,0,3,
%U 0,5,0,3,0,6,0,1,0,3,0,2,0,1,0,1,1,4,0
%N Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.
%C The number of submultisets of a partition is the product of its multiplicities, each plus one.
%H Alois P. Heinz, <a href="/A325987/b325987.txt">Rows n = 0..60, flattened</a>
%F Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - _Alois P. Heinz_, Aug 17 2019
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 0 2
%e 0 1 1 1 1 1
%e 0 1 0 2 0 3 0 1
%e 0 1 1 3 0 1 1 2 1 1
%e 0 1 0 3 0 3 0 4 0 1 0 3
%e 0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
%e 0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
%e 0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
%e Row n = 7 counts the following partitions (empty columns not shown):
%e (7) (43) (322) (421) (31111) (3211)
%e (52) (331) (2221) (22111)
%e (61) (511) (4111) (211111)
%e (1111111)
%t Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==k&]],{n,0,10},{k,1,Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]
%Y Row lengths are A088881.
%Y Row sums are A000041.
%Y Diagonal n = k is A325830 interspersed with zeros.
%Y Diagonal n + 1 = k is A325828.
%Y Diagonal n - 1 = k is A325836.
%Y Column k = 3 appears to be A137719.
%Y Cf. A000005, A000712, A002033, A005179, A088880, A108917, A126796.
%Y Cf. A325694, A325792, A325793, A325831, A325832, A325833, A325834.
%K nonn,look,tabf
%O 0,10
%A _Gus Wiseman_, May 30 2019