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%I #23 Jan 12 2020 23:57:09
%S 0,1,3,7,14,27,49,86,146,242,392,623,973,1498,2274,3411,5059,7427,
%T 10801,15572,22267,31602,44533,62338,86716,119918,164903,225566,
%U 306993,415814,560641,752622,1006132,1339677,1776980,2348384,3092594,4058848,5309608,6923959
%N Number of partitions of 2n+1 of type OO (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="/A236914/b236914.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%e The partitions of 5 of type OO are [4,1], [3,2], [2,1,1,1], so that a(2) = 3.
%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+1$2)[4]:
%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*) (* _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+1, 2*n+1][[4]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 27 2015, after _Alois P. Heinz_ *)
%Y Cf. A000041, A000701, A027187, A027193, A160786, A236559, A236913.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Feb 01 2014
%E More terms from _Alois P. Heinz_, Feb 16 2014
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