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A241312 Number of partitions p of n into distinct parts, including floor(mean(p)). 16

%I

%S 0,1,1,2,1,2,2,3,2,4,4,5,5,7,7,11,10,13,15,19,19,25,28,34,37,44,49,61,

%T 66,80,87,102,114,134,156,174,189,221,252,294,321,369,404,461,521,586,

%U 663,759,822,918,1021,1156,1305,1472,1621,1803,1949,2202,2469

%N Number of partitions p of n into distinct parts, including floor(mean(p)).

%F a(n) + A241313(n) = A000009(n) for n >= 1.

%e a(10) counts these 4 partitions: {10}, {5,4,1}, {5,3,2}, {4,3,2,1}.

%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]]]], {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. A241313, A241314, A241315, A241318, A000009.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Apr 19 2014

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Last modified June 1 01:59 EDT 2020. Contains 334758 sequences. (Running on oeis4.)