A241445 - OEIS

%I #4 Apr 28 2014 17:55:10

%S 1,0,1,1,3,2,4,4,7,7,11,15,21,24,36,45,59,73,99,114,155,183,241,287,

%T 371,433,567,668,842,1003,1270,1483,1856,2205,2707,3210,3940,4627,

%U 5661,6656,8050,9489,11432,13385,16070,18855,22459,26310,31253,36487,43249

%N Number of partitions of n such that the number of parts having multiplicity 1 is not a part and the number of distinct parts is not a part.

%F a(n) + A241446(n) = A000041(n) for n >= 0.

%e a(6) counts these 4 partitions:  6, 51, 33, 222.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241442 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, d[p]] ], {n, 0,  z}] (* A241443 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241444 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241445 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, d[p]] ], {n, 0, z}] (* A241446 *)

%Y Cf. A241442, A241443, A241444, A241446, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 23 2014