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Number of partitions p of n such that the number of parts is a part and max(p) - min(p) is a part.
5

%I #4 Apr 27 2014 10:24:19

%S 0,0,0,1,0,0,2,0,1,2,3,3,6,4,10,8,12,12,20,17,29,28,45,48,68,69,98,

%T 103,134,148,194,208,271,298,377,424,528,589,735,825,1004,1139,1381,

%U 1551,1874,2116,2528,2869,3401,3848,4559,5165,6066,6891,8060,9136

%N Number of partitions p of n such that the number of parts is a part and max(p) - min(p) is a part.

%F a(n) + A241383(n) + A241384(n) = A241386(n) for n >= 0.

%e a(9) counts these 2 partitions: 432, 4311.

%t z = 40; f[n_] := f[n] = IntegerPartitions[n];

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241382 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241383 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && ! MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241384 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && ! MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241385 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] || MemberQ[p, Max[p] - Min[p]]], {n, 0, z}] (* A241386 *)

%Y Cf. A241383, A241384, A241385, A241386.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 21 2014