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

%I #4 Apr 27 2014 10:24:31

%S 0,0,0,0,1,2,2,4,6,8,11,12,21,24,33,42,57,68,95,110,147,176,223,262,

%T 344,402,508,607,758,894,1117,1309,1614,1905,2315,2722,3306,3870,4657,

%U 5468,6536,7642,9113,10635,12608,14716,17346,20197,23770,27597,32334

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

%F a(n) + A241382(n) + A241384(n) = A241386(n) for n >= 0.

%e a(9) counts these 8 partitions: 63, 3321, 32211, 321111, 22221, 222111, 221111, 2111111.

%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. A241382, A241384, A241385, A241386.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 21 2014