A239239 Number of strict partitions of n having fewer odd parts than even.

0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 4, 4, 7, 5, 11, 7, 16, 10, 23, 15, 32, 21, 43, 32, 57, 45, 74, 66, 96, 92, 123, 129, 157, 175, 199, 239, 253, 316, 320, 419, 406, 544, 514, 704, 652, 898, 825, 1142, 1045, 1435, 1321, 1798, 1669, 2234, 2103, 2766, 2646, 3404

Offset: 0

Author

Clark Kimberling, Mar 13 2014

Keywords

• nonn
• easy

Comments

a(n) = Sum_{k<=-1} A240021(n,k). - Alois P. Heinz, Apr 02 2014

Examples

			a(6) counts these partitions:  6, 42.
		

Links

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

Crossrefs

Cf. A239240, A239241, A239242, A239243, A000009.

Programs

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 29 2016, after Alois P. Heinz *)

Formula

a(n) + A239243(n) = A000009(n) for n >=1.