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

0,3

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.

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A274233 A369576 A256209 * A152902 A027084 A014172

Adjacent sequences: A236911 A236912 A236913 * A236915 A236916 A236917

Keyword

nonn,easy

Author

Clark Kimberling, Feb 01 2014

Extensions

More terms from Alois P. Heinz, Feb 16 2014

Status

approved