A240863 Number of partitions of n into distinct parts of which the number of odd parts is a part.

0, 1, 0, 1, 0, 1, 1, 2, 1, 3, 3, 5, 4, 7, 7, 11, 10, 15, 15, 22, 22, 31, 31, 42, 43, 58, 59, 78, 82, 105, 109, 139, 146, 183, 193, 239, 255, 311, 331, 402, 430, 516, 553, 659, 710, 839, 904, 1061, 1146, 1337, 1446, 1679, 1819, 2099, 2276, 2615, 2838, 3246

Offset

0,8

Links

Table of n, a(n) for n=0..57.

Formula

a(n) + A240870(n) = A000009(n) for n >= 0.

Example

a(10) counts these 3 partitions:  721, 532, 4321.

Mathematica

 z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
    t1 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]]], {n, 0, z}] (* A240862 *)
    t2 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240863, *)
    t3 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240864 *)
    t4 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] || MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240865 *)
    t5 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240866 *)
    t6 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240867 *)
    t7 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240868 *)

Crossrefs

Cf. A240862, A240864, A240865, A240866, A240867, A204868; for analogous sequences for unrestricted partitions, see A240573-A240579.

Sequence in context: A178133 A343379 A026927 * A288005 A237832 A074500

Adjacent sequences: A240860 A240861 A240862 * A240864 A240865 A240866

Keyword

nonn,easy

Author

Clark Kimberling, Apr 14 2014

Status

approved