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A236559 Number of partitions of 2n of type EO (see Comments).

%I A236559 #32 Jan 12 2020 11:29:10
%S A236559 0,1,2,5,10,20,37,66,113,190,310,497,782,1212,1851,2793,4163,6142,
%T A236559 8972,12989,18646,26561,37556,52743,73593,102064,140736,193011,263333,
%U A236559 357521,483129,649960,870677,1161604,1543687,2043780,2696156,3544485,4644241,6065739
%N A236559 Number of partitions of 2n of type EO (see Comments).
%C A236559 The partitions of n are partitioned into four types:
%C A236559 EO, even # of odd parts and odd  # of even parts, A236559;
%C A236559 OE, odd  # of odd parts and even # of even parts, A160786;
%C A236559 EE, even # of odd parts and even # of even parts, A236913;
%C A236559 OO, odd  # of odd parts and odd  # of even parts, A236914.
%C A236559 A236559 and A160786 are the bisections of A027193;
%C A236559 A236913 and A236914 are the bisections of A027187.
%H A236559 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 A236559 The partitions of 4 of type EO are [4] and [2,1,1], so that a(2) = 2.
%e A236559 type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
%e A236559 EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
%e A236559 OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
%e A236559 EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
%e A236559 OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27
%p A236559 b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
%p A236559       `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
%p A236559       `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
%p A236559           [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
%p A236559     end:
%p A236559 a:= n-> b(2*n$2)[3]:
%p A236559 seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 16 2014
%t A236559 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 A236559 OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
%t A236559 OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
%t A236559 m1 (* A236559, type EO*)
%t A236559 m2 (* A160786, type OE*)
%t A236559 m3 (* A236913, type EE*)
%t A236559 m4 (* A236914, type OO*)
%t A236559 (* _Peter J. C. Moses_, Feb 03 2014 *)
%t A236559 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 A236559 Cf. A000041, A000701, A027187, A027193, A160786, A236913, A236914.
%K A236559 nonn,easy
%O A236559 0,3
%A A236559 _Clark Kimberling_, Feb 01 2014
%E A236559 More terms from and definition corrected by _Alois P. Heinz_, Feb 16 2014