This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336141 #17 May 20 2021 04:25:13
%S A336141 1,1,2,5,9,17,41,71,138,270,518,938,1863,3323,6163,11436,20883,37413,
%T A336141 69257,122784,221873,397258,708142,1249955,2236499,3917628,6909676,
%U A336141 12130972,21251742,36973609,64788378,112103360,194628113,336713377,581527210,1000153063
%N A336141 Number of ways to choose a strict composition of each part of an integer partition of n.
%C A336141 A strict composition of n is a finite sequence of distinct positive integers summing to n.
%H A336141 Alois P. Heinz, <a href="/A336141/b336141.txt">Table of n, a(n) for n = 0..5623</a>
%F A336141 G.f.: Product_{k >= 1} 1/(1 - A032020(k)*x^k).
%e A336141 The a(1) = 1 through a(5) = 17 ways:
%e A336141 (1) (2) (3) (4) (5)
%e A336141 (1),(1) (1,2) (1,3) (1,4)
%e A336141 (2,1) (3,1) (2,3)
%e A336141 (2),(1) (2),(2) (3,2)
%e A336141 (1),(1),(1) (3),(1) (4,1)
%e A336141 (1,2),(1) (3),(2)
%e A336141 (2,1),(1) (4),(1)
%e A336141 (2),(1),(1) (1,2),(2)
%e A336141 (1),(1),(1),(1) (1,3),(1)
%e A336141 (2,1),(2)
%e A336141 (3,1),(1)
%e A336141 (2),(2),(1)
%e A336141 (3),(1),(1)
%e A336141 (1,2),(1),(1)
%e A336141 (2,1),(1),(1)
%e A336141 (2),(1),(1),(1)
%e A336141 (1),(1),(1),(1),(1)
%p A336141 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A336141 `if`(n=0, p!, b(n, i-1, p)+b(n-i, min(n-i, i-1), p+1)))
%p A336141 end:
%p A336141 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A336141 g(n, i-1)+b(i$2, 0)*g(n-i, min(n-i, i)))
%p A336141 end:
%p A336141 a:= n-> g(n$2):
%p A336141 seq(a(n), n=0..38); # _Alois P. Heinz_, Jul 31 2020
%t A336141 Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
%t A336141 (* Second program: *)
%t A336141 b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
%t A336141 If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
%t A336141 g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1] +
%t A336141 b[i, i, 0] g[n-i, Min[n-i, i]]];
%t A336141 a[n_] := g[n, n];
%t A336141 a /@ Range[0, 38] (* _Jean-François Alcover_, May 20 2021, after _Alois P. Heinz_ *)
%Y A336141 Multiset partitions of partitions are A001970.
%Y A336141 Strict compositions are counted by A032020, A072574, and A072575.
%Y A336141 Splittings of partitions are A323583.
%Y A336141 Splittings of partitions with distinct sums are A336131.
%Y A336141 Cf. A008289, A011782, A304786, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
%Y A336141 Partitions:
%Y A336141 - Partitions of each part of a partition are A063834.
%Y A336141 - Compositions of each part of a partition are A075900.
%Y A336141 - Strict partitions of each part of a partition are A270995.
%Y A336141 - Strict compositions of each part of a partition are A336141.
%Y A336141 Strict partitions:
%Y A336141 - Partitions of each part of a strict partition are A271619.
%Y A336141 - Compositions of each part of a strict partition are A304961.
%Y A336141 - Strict partitions of each part of a strict partition are A279785.
%Y A336141 - Strict compositions of each part of a strict partition are A336142.
%Y A336141 Compositions:
%Y A336141 - Partitions of each part of a composition are A055887.
%Y A336141 - Compositions of each part of a composition are A133494.
%Y A336141 - Strict partitions of each part of a composition are A304969.
%Y A336141 - Strict compositions of each part of a composition are A307068.
%Y A336141 Strict compositions:
%Y A336141 - Partitions of each part of a strict composition are A336342.
%Y A336141 - Compositions of each part of a strict composition are A336127.
%Y A336141 - Strict partitions of each part of a strict composition are A336343.
%Y A336141 - Strict compositions of each part of a strict composition are A336139.
%K A336141 nonn
%O A336141 0,3
%A A336141 _Gus Wiseman_, Jul 18 2020