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 A363224 #10 Jul 06 2024 21:09:59
%S A363224 0,1,1,5,8,21,44,94,197,416,857,1766,3621,7392,15032,30493,61708,
%T A363224 124646,251359,506203,1018279,2046454,4109534,8246985,16540791,
%U A363224 33160051,66451484,133122753,266612828,533839069,1068701695,2139110054,4281063708,8566862025
%N A363224 Number of integer compositions of n in which the least part appears more than once.
%C A363224 Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.
%H A363224 John Tyler Rascoe, <a href="/A363224/b363224.txt">Table of n, a(n) for n = 1..1000</a>
%F A363224 G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - _John Tyler Rascoe_, Jul 06 2024
%e A363224 The a(1) = 0 through a(6) = 21 compositions:
%e A363224 . (11) (111) (22) (113) (33)
%e A363224 (112) (131) (114)
%e A363224 (121) (311) (141)
%e A363224 (211) (1112) (222)
%e A363224 (1111) (1121) (411)
%e A363224 (1211) (1113)
%e A363224 (2111) (1122)
%e A363224 (11111) (1131)
%e A363224 (1212)
%e A363224 (1221)
%e A363224 (1311)
%e A363224 (2112)
%e A363224 (2121)
%e A363224 (2211)
%e A363224 (3111)
%e A363224 (11112)
%e A363224 (11121)
%e A363224 (11211)
%e A363224 (12111)
%e A363224 (21111)
%e A363224 (111111)
%t A363224 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Min@@#]>1&]],{n,15}]
%o A363224 (PARI)
%o A363224 C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
%o A363224 C_x(35) \\ _John Tyler Rascoe_, Jul 06 2024
%Y A363224 The complement is counted by A105039.
%Y A363224 For partitions instead of compositions we have A117989.
%Y A363224 Row sums of columns k > 1 of A238342.
%Y A363224 If all parts appear more than once we have A240085, for partitions A007690.
%Y A363224 If the least part appears exactly twice we have A241862.
%Y A363224 For greatest instead of least we have A363262, see triangle A238341.
%Y A363224 A000041 counts integer partitions, strict A000009.
%Y A363224 A032020 counts strict compositions.
%Y A363224 A067029 gives last exponent in prime factorization, first A071178.
%Y A363224 A261982 counts compositions with some part appearing more than once.
%Y A363224 A362607 counts partitions with multiple modes, co-modes A362609.
%Y A363224 A362608 counts partitions with a unique mode, co-mode A362610.
%Y A363224 Cf. A002865, A008284, A097979, A243737.
%K A363224 nonn
%O A363224 1,4
%A A363224 _Gus Wiseman_, Jun 04 2023