login
Number of partitions p of n such that the number of distinct parts is not a part and max(p) - min(p) is not a part.
5

%I #4 Apr 27 2014 10:25:40

%S 1,0,1,1,3,3,6,8,12,15,22,31,37,56,67,92,116,156,190,255,310,406,498,

%T 638,787,988,1212,1517,1856,2290,2802,3441,4158,5099,6166,7460,9015,

%U 10879,13049,15716,18752,22469,26798,31961,37890,45148,53376,63253,74626

%N Number of partitions p of n such that the number of distinct parts is not a part and max(p) - min(p) is not a part.

%F a(n) + A241391(n) = A000041(n) for n >= 0.

%e a(6) counts these 6 partitions: 6, 51, 411, 33, 3111, 222.

%t z = 40; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]];

%t Table[Count[f[n], p_ /; MemberQ[p, d[p]] && MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241387 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, d[p]] && MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241388 *)

%t Table[Count[f[n], p_ /; MemberQ[p, d[p]] && ! MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241389 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, d[p]] && ! MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241390 *)

%t Table[Count[f[n], p_ /; MemberQ[p, d[p]] || MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241391 *)

%Y Cf. A241387, A241388, A241389, A241391, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 21 2014