A236559 - OEIS

A236559

Number of partitions of 2n of type EO (see Comments).

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

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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-François 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