A241451 - OEIS

%I #5 Apr 28 2014 18:47:14

%S 0,1,0,1,1,4,6,9,13,21,28,35,52,68,89,121,155,205,264,340,433,555,693,

%T 872,1095,1367,1695,2107,2580,3180,3911,4773,5803,7083,8565,10364,

%U 12515,15077,18075,21721,25936,31023,36954,43984,52152,61966,73238,86586

%N Number of partitions p of n such that the number of parts having multiplicity 1 is a part or max(p) - min(p) is a part.

%F a(n) + A241450(n) = A000041(n) for n >= 0.

%e a(6) counts these 6 partitions:  42, 411, 321, 3111, 2211, 21111.

%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, Max[p]-Min[p]]], {n, 0, z}]  (* A241447 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, Max[p]-Min[p]] ], {n, 0,  z}] (* A241448 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, Max[p]-Min[p]] ], {n, 0, z}] (* A241449 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, Max[p]-Min[p]] ], {n, 0, z}] (* A241450 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, Max[p]-Min[p]] ], {n, 0, z}] (* A241451 *)

%Y Cf. A241447, A241448, A241449, A241450.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 23 2014