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Number of partitions of n containing m(1) as a part, where m denotes multiplicity.
6

%I #8 Apr 29 2014 02:57:01

%S 0,1,0,1,2,2,4,5,8,10,16,19,29,36,51,63,89,108,148,182,242,297,390,

%T 475,615,750,955,1161,1466,1774,2217,2679,3316,3994,4911,5892,7197,

%U 8613,10451,12470,15055,17905,21508,25513,30503,36081,42966,50678,60117,70732

%N Number of partitions of n containing m(1) as a part, where m denotes multiplicity.

%e a(6) counts these 5 partitions: 51, 421, 331, 3211, 2221.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n];

%t Table[Count[f[n], p_ /; MemberQ[p, Count[p, 1]]], {n, 0, z}] (* A240486 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Count[p, 2]]], {n, 0, z}] (* A240487 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Count[p, 3]]], {n, 0, z}] (* A240488 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Count[p, 4]]], {n, 0, z}] (* A240489 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Count[p, 5]]], {n, 0, z}] (* A240490 *)

%Y Cf. A240487 - A240490.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 06 2014