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A239833
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Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.
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8
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0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 17, 22, 28, 36, 46, 58, 72, 92, 113, 141, 174, 216, 263, 324, 394, 481, 583, 707, 852, 1029, 1235, 1481, 1774, 2118, 2524, 3003, 3567, 4225, 5003, 5906, 6968, 8202, 9646, 11317, 13275, 15531, 18160, 21195, 24718, 28772
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OFFSET
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0,5
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LINKS
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FORMULA
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EXAMPLE
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a(10) counts these 10 partitions: [10], [1,8,1], [7,2,1], [3,6,1], [5,4,1], [5,3,2], [3,4,3], [4,1,2,1,2], [2,3,2,1,2], [1,2,1,2,1,2,1].
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MAPLE
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b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
end:
a:= n-> b(n$2, -1) +b(n$2, 1):
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MATHEMATICA
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p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, _?OddQ] - Count[#, _?EvenQ]] == 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions*)
t = Table[Length[p[n]], {n, 0, 60}] (* A239833 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]>n, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, -1] + b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 12 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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STATUS
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approved
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