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%I #7 Jun 06 2023 08:17:57
%S 0,1,1,2,4,9,18,37,73,145,287,570,1134,2264,4526,9061,18152,36374,
%T 72884,146011,292416,585422,1171632,2344136,4688821,9376832,18749169,
%U 37485358,74939850,149813328,299492966,598729533,1196987066,2393137399,4784846896,9567357951
%N Number of integer compositions of n in which the greatest part appears more than once.
%C Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.
%e The a(2) = 1 through a(6) = 9 compositions:
%e (11) (111) (22) (122) (33)
%e (1111) (212) (222)
%e (221) (1122)
%e (11111) (1212)
%e (1221)
%e (2112)
%e (2121)
%e (2211)
%e (111111)
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Max@@#]>1&]],{n,15}]
%Y For partitions instead of compositions we have A002865.
%Y The complement is counted by A097979 shifted left.
%Y Row sums of columns k > 1 of A238341.
%Y If all parts appear more than once we have A240085, for partitions A007690.
%Y If the greatest part appears exactly twice we have A243737.
%Y For least instead of greatest we have A363224, see triangle A238342.
%Y A000041 counts integer partitions, strict A000009.
%Y A032020 counts strict compositions.
%Y A067029 gives last exponent in prime factorization, first A071178.
%Y A261982 counts compositions with some part appearing more than once.
%Y Cf. A008284, A105039, A117989, A362607, A362608, A362612, A362614.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jun 04 2023
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