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Number of partitions p of n into distinct parts, not including floor(mean(p)).
6

%I #6 Nov 22 2021 09:59:05

%S 0,0,0,0,1,1,2,2,4,4,6,7,10,11,15,16,22,25,31,35,45,51,61,70,85,98,

%T 116,131,156,176,209,238,276,314,356,411,479,539,612,688,792,891,1022,

%U 1149,1295,1462,1641,1831,2088,2346,2637,2941,3277,3648,4097,4575

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

%F a(n) + A241312(n) = A000009(n) for n >= 1.

%e a(10) counts these 6 partitions: 91, 82, 73, 721, 64, 631.

%t z = 30;

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

%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241312 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241313 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241314 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241315 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241316 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Round[Mean[p]]]], {n, 0, ] (* A241317 *)

%Y Cf. A241312, A241314, A241315, A241318, A000009.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 19 2014