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A241377
Number of partitions of n such that the number of parts is a part and the number of distinct parts is a part.
5
0, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 6, 8, 11, 12, 16, 20, 27, 31, 40, 49, 59, 72, 95, 110, 133, 164, 196, 237, 289, 351, 410, 502, 595, 704, 843, 1009, 1193, 1422, 1658, 1983, 2332, 2744, 3204, 3796, 4459, 5189, 6083, 7116, 8292, 9677, 11222, 13041, 15235
OFFSET
0,7
FORMULA
a(n) + A241378(n) + A241379(n) = A241381(n) for n >= 0.
EXAMPLE
a(9) counts these 4 partitions: 72, 531, 432, 4311.
MATHEMATICA
z = 70; 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