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Number of partitions p of n into distinct parts, including floor(mean(p)) but not ceiling(mean(p)).
5

%I #7 Jun 18 2015 04:00:20

%S 0,0,0,0,0,0,0,1,1,0,2,2,0,4,4,2,4,8,5,12,5,10,19,22,8,21,34,34,29,54,

%T 21,72,57,83,109,70,59,158,179,195,99,259,160,332,317,270,471,537,286,

%U 542,537,816,852,1051,751,995,828,1584,1789,2005,941,2431

%N Number of partitions p of n into distinct parts, including floor(mean(p)) but not ceiling(mean(p)).

%e a(11) counts these 2 partitions: 641, 542.

%t z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];

%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241318 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241319 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241320 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241321 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241322 *)

%Y Cf. A241318, A241319, A241321, A241322, A241312, A000009.

%K nonn,easy

%O 0,11

%A _Clark Kimberling_, Apr 19 2014