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A241381
Number of partitions of n such that the number of parts or the number of distinct parts is a part.
5
0, 1, 1, 2, 3, 5, 5, 9, 11, 17, 22, 30, 41, 53, 73, 92, 121, 155, 200, 255, 324, 408, 516, 643, 796, 1009, 1231, 1529, 1872, 2317, 2792, 3452, 4168, 5073, 6115, 7433, 8875, 10741, 12816, 15400, 18344, 21923, 25997, 30999, 36693, 43412, 51334, 60629, 71339
OFFSET
0,4
FORMULA
a(n) + A241380(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 5 partitions: 41, 321, 2211, 21111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := [p] = Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241377 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241378 *)
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}] (* A241379 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}] (* A241380 *)
Table[Count[f[n], p_ /; MemberQ[p, Length[p]] || MemberQ[p, d[p]]], {n, 0, z}] (* A241381 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 21 2014
STATUS
approved