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Number of partitions of n such that m(1) > m(3), where m = multiplicity.
2

%I #6 Apr 06 2014 04:17:25

%S 0,1,1,2,2,5,6,10,12,20,25,37,46,67,84,116,145,197,246,325,404,527,

%T 653,837,1032,1310,1609,2018,2467,3070,3738,4612,5591,6854,8277,10080,

%U 12125,14688,17604,21212,25333,30389,36172,43201,51256,60981,72132,85498

%N Number of partitions of n such that m(1) > m(3), where m = multiplicity.

%F a(n) + A182714(n) + A240058(n) = A000041(n) for n >= 0.

%e a(6) counts these 6 partitions: 51, 411, 3111, 2211, 21111, 111111.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}] (* A182714 *)

%t t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 3]], {n, 0, z}] (* A182714(n+3) *)

%t t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 3]], {n, 0, z}] (* A240058 *)

%t t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 3]], {n, 0, z}] (* A240059 *)

%t t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 3]], {n, 0, z}] (* A240059(n+1) *)

%Y Cf. A182714, A240058, A000041.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Mar 31 2014