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A241446
Number of partitions of n such that the number of parts having multiplicity 1 is a part or the number of distinct parts is a part.
5
0, 1, 1, 2, 2, 5, 7, 11, 15, 23, 31, 41, 56, 77, 99, 131, 172, 224, 286, 376, 472, 609, 761, 968, 1204, 1525, 1869, 2342, 2876, 3562, 4334, 5359, 6493, 7938, 9603, 11673, 14037, 17010, 20354, 24529, 29288, 35094, 41742, 49876, 59105, 70279, 83099, 98444
OFFSET
0,4
FORMULA
a(n) + A241445(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 7 partitions: 42, 411, 321, 3111, 2211, 21111, 111111.
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