login
A241379
Number of partitions of n such that the number of parts is a part and the number of distinct parts is not a part.
5
0, 0, 0, 0, 1, 1, 0, 2, 1, 3, 3, 6, 6, 10, 12, 18, 21, 28, 35, 48, 56, 78, 93, 115, 143, 187, 219, 282, 337, 419, 496, 629, 736, 912, 1090, 1324, 1564, 1901, 2238, 2720, 3187, 3821, 4501, 5387, 6291, 7455, 8770, 10341, 12080, 14227, 16575, 19479, 22676
OFFSET
0,8
FORMULA
a(n) + A241377(n) + A241378(n) = A241381(n) for n >= 0.
EXAMPLE
a(9) counts these 3 partitions: 51111, 4221, 333.
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