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A236559 Number of partitions of 2n of type EO (see Comments). 38
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-François Alcover, Oct 27 2015, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A327289 A327288 A102688 * A275388 A341581 A001629
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)