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A241442
Number of partitions of n such that the number of parts having multiplicity 1 is a part and the number of distinct parts is a part.
5
0, 1, 0, 1, 1, 3, 3, 4, 5, 8, 9, 12, 17, 25, 30, 43, 52, 73, 93, 119, 147, 191, 238, 303, 370, 473, 573, 721, 873, 1089, 1326, 1640, 1954, 2438, 2900, 3556, 4240, 5181, 6140, 7452, 8851, 10626, 12600, 15090, 17812, 21248, 25063, 29686, 34969, 41344, 48465
OFFSET
0,6
FORMULA
a(n) + A241443(n) + A241444(n) = A241446(n) for n >= 0.
EXAMPLE
a(6) counts these 3 partitions: 42, 321, 21111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241442 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241443 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241444 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241445 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, d[p]] ], {n, 0, z}] (* A241446 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 23 2014
STATUS
approved