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A236914
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Number of partitions of 2n+1 of type OO (see Comments).
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36
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0, 1, 3, 7, 14, 27, 49, 86, 146, 242, 392, 623, 973, 1498, 2274, 3411, 5059, 7427, 10801, 15572, 22267, 31602, 44533, 62338, 86716, 119918, 164903, 225566, 306993, 415814, 560641, 752622, 1006132, 1339677, 1776980, 2348384, 3092594, 4058848, 5309608, 6923959
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The partitions of 5 of type OO are [4,1], [3,2], [2,1,1,1], so that a(2) = 3.
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
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MAPLE
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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+1$2)[4]:
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MATHEMATICA
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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]];
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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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