%I #11 Dec 15 2022 17:43:29
%S 1,1,2,8,18,54,156,412,1168,3200,8848,24192,66632,181912,495536,
%T 1354880,3680352,9997056,27093216,73376512,198355840,535319168,
%U 1443042688,3884515008,10445579840,28046885824,75225974912,201536064896,539339293824,1441781213952
%N Number of finite sequences of distinct integer compositions with total sum n.
%H Alois P. Heinz, <a href="/A358907/b358907.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(1) = 1 through a(4) = 18 sequences:
%e ((1)) ((2)) ((3)) ((4))
%e ((11)) ((12)) ((13))
%e ((21)) ((22))
%e ((111)) ((31))
%e ((1)(2)) ((112))
%e ((2)(1)) ((121))
%e ((1)(11)) ((211))
%e ((11)(1)) ((1111))
%e ((1)(3))
%e ((3)(1))
%e ((1)(12))
%e ((11)(2))
%e ((1)(21))
%e ((12)(1))
%e ((2)(11))
%e ((21)(1))
%e ((1)(111))
%e ((111)(1))
%p g:= proc(n) option remember; ceil(2^(n-1)) end:
%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
%p add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..32); # _Alois P. Heinz_, Dec 15 2022
%t comps[n_]:=Join@@Permutations/@IntegerPartitions[n];
%t Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]
%Y For sets instead of sequences we have A098407, partitions A261049.
%Y This is the strict case of A133494.
%Y The case of distinct sums is A336127, constant sums A074854.
%Y The version for sequences of partitions is A358906.
%Y A001970 counts multiset partitions of integer partitions.
%Y A063834 counts twice-partitions.
%Y A218482 counts sequences of compositions with weakly decreasing lengths.
%Y A358830 counts twice-partitions with distinct lengths.
%Y A358901 counts partitions with all different Omegas.
%Y A358914 counts twice-partitions into distinct strict partitions.
%Y Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 07 2022
%E a(16)-a(29) from _Alois P. Heinz_, Dec 15 2022