A236914 Number of partitions of 2n+1 of type OO (see Comments).

0, 1, 3, 7, 14, 27, 49, 86, 146, 242, 392, 623, 973, 1498, 2274, 3411, 5059, 7427, 10801, 15572, 22267, 31602, 44533, 62338, 86716, 119918, 164903, 225566, 306993, 415814, 560641, 752622, 1006132, 1339677, 1776980, 2348384, 3092594, 4058848, 5309608, 6923959

Offset: 0

Author

Clark Kimberling, Feb 01 2014

Keywords

• nonn
• easy

Comments

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

Examples

			The partitions of 5 of type OO are [4,1], [3,2], [2,1,1,1], so that a(2) = 3.
type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A000041, A000701, A027187, A027193, A160786, A236559, A236913.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n+1$2)[4]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

Mathematica

z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,  OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,       OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
m1 (* A236559, type EO*)
m2 (* A160786, type OE*)
m3 (* A236913, type EE*)
m4 (* A236914, type OO*) (* Peter J. C. Moses, Feb 03 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[i > n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2] == 0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n+1, 2*n+1][[4]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 16 2014