A236559 - OEIS

%I #32 Jan 12 2020 11:29:10

%S 0,1,2,5,10,20,37,66,113,190,310,497,782,1212,1851,2793,4163,6142,

%T 8972,12989,18646,26561,37556,52743,73593,102064,140736,193011,263333,

%U 357521,483129,649960,870677,1161604,1543687,2043780,2696156,3544485,4644241,6065739

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

%C The partitions of n are partitioned into four types:

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

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

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

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

%C A236559 and A160786 are the bisections of A027193;

%C A236913 and A236914 are the bisections of A027187.

%H Seiichi Manyama, <a href="/A236559/b236559.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%e The partitions of 4 of type EO are [4] and [2,1,1], so that a(2) = 2.

%e type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11

%e EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0

%e OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29

%e EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0

%e OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],

%p       `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->

%p       `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],

%p           [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))

%p     end:

%p a:= n-> b(2*n$2)[3]:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 16 2014

%t 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]} &,

%t OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,

%t OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];

%t m1 (* A236559, type EO*)

%t m2 (* A160786, type OE*)

%t m3 (* A236913, type EE*)

%t m4 (* A236914, type OO*)

%t (* _Peter J. C. Moses_, Feb 03 2014 *)

%t 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_ *)

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

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Feb 01 2014

%E More terms from and definition corrected by _Alois P. Heinz_, Feb 16 2014