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Number of partitions of n such that m(greatest part) >= m(1), where m = multiplicity.
5

%I #4 Apr 06 2014 04:18:43

%S 1,1,2,3,4,5,8,9,14,17,24,29,42,49,68,83,110,133,176,211,274,331,420,

%T 507,640,767,956,1149,1416,1695,2078,2477,3014,3589,4334,5147,6188,

%U 7321,8756,10341,12306,14491,17182,20175,23828,27919,32848,38393,45038,52505

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

%F a(n) = A240078(n) + A117995(n) for n >= 0.

%e a(7) counts these 9 partitions: 7, 61, 52, 43, 421, 331, 322, 2221, 1111111.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, Max[p]] < Count[p, 1]], {n, 0, z}] (* A240076 *)

%t t2 = Table[Count[f[n], p_ /; Count[p, Max[p]] <= Count[p, 1]], {n, 0, z}] (* A240077 *)

%t t3 = Table[Count[f[n], p_ /; Count[p, Max[p]] == Count[p, 1]], {n, 0, z}] (* A240078 *)

%t t4 = Table[Count[f[n], p_ /; Count[p, Max[p]] > Count[p, 1]], {n, 0, z}] (* A117995 *)

%t t5 = Table[Count[f[n], p_ /; Count[p, Max[p]] >= Count[p, 1]], {n, 0, z}] (* A240080 *)

%Y Cf. A240076, A240077, A240078, A117995.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 01 2014