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

0,7

Comments

a(n) = Sum_{k<=-1} 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) + A239243(n) = A000009(n) for n >=1.

Example

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

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 *)

Crossrefs

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

Sequence in context: A246583 A244788 A078660 * A241701 A347662 A060177

Adjacent sequences: A239236 A239237 A239238 * A239240 A239241 A239242

Keyword

nonn,easy

Author

Clark Kimberling, Mar 13 2014

Status

approved