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A236559 Number of partitions of 2n of type EO (see Comments). 4
0, 1, 2, 5, 10, 20, 37, 66, 113, 190, 310, 497, 782, 1212, 1851, 2793, 4163, 6142, 8972, 12989, 18646, 26561, 37556, 52743, 73593, 102064, 140736, 193011, 263333, 357521, 483129, 649960, 870677, 1161604, 1543687, 2043780, 2696156, 3544485, 4644241, 6065739 (list; graph; refs; listen; history; text; internal format)
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 4 of type EO are [4] and [2,1,1], so that a(2) = 2.

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$2)[3]:

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, 2*n][[3]]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Oct 27 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A000701, A027187, A027193, A160786, A236913, A236914.

Sequence in context: A327289 A327288 A102688 * A275388 A341581 A001629

Adjacent sequences:  A236556 A236557 A236558 * A236560 A236561 A236562

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 01 2014

EXTENSIONS

More terms from and definition corrected by Alois P. Heinz, Feb 16 2014

STATUS

approved

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Last modified April 21 17:36 EDT 2021. Contains 343156 sequences. (Running on oeis4.)