A239240 Number of partitions of n into distinct parts for which (number of odd parts) <= (number of even parts).

1, 0, 1, 1, 1, 2, 2, 4, 2, 6, 4, 9, 6, 13, 10, 18, 15, 24, 24, 32, 35, 43, 51, 56, 72, 74, 100, 97, 136, 128, 183, 168, 241, 222, 315, 290, 408, 381, 522, 497, 664, 647, 839, 837, 1054, 1081, 1317, 1384, 1641, 1767, 2035, 2242, 2519, 2831, 3108, 3555, 3828

Offset

0,6

Comments

a(n) = Sum_{k<=0} A240021(n,k). - Alois P. Heinz, Apr 02 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) + A239242(n) = A000009(n) for n >=1.

Example

a(7) = 4 counts these partitions:  61, 52, 43, 421.

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t<=0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

Mathematica

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] < Count[#, _?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] <= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] == Count[#, _?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] > Count[#, _?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] >= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t <= 0, 1, 0], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Crossrefs

Cf. A239239, A239241, A239242, A239243, A000009.

Sequence in context: A028913 A185048 A240828 * A054929 A236628 A078429

Adjacent sequences: A239237 A239238 A239239 * A239241 A239242 A239243

Keyword

nonn,easy

Author

Clark Kimberling, Mar 13 2014

Status

approved