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A032189
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Number of ways to partition n elements into pie slices each with an odd number of elements.
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0
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1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 19, 30, 41, 63, 94, 142, 211, 328, 493, 765, 1170, 1810, 2787, 4340, 6713, 10461, 16274, 25414, 39651, 62074, 97109, 152287, 238838, 375166, 589527, 927554, 1459961, 2300347, 3626242, 5721044, 9030451, 14264308, 22542397, 35646311, 56393862, 89264834, 141358275
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OFFSET
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1,3
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COMMENTS
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a(n) is also the total number of cyclic compositions of n into odd parts assuming that two compositions are equivalent if one can be obtained from the other by a cyclic shift. For example, a(5)=3 because 5 has the following three cyclic compositions into odd parts: 5, 1+3+1, 1+1+1+1+1. - Petros Hadjicostas, Dec 27 2016
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LINKS
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FORMULA
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"CIK" (necklace, indistinct, unlabeled) transform of 1, 0, 1, 0...(odds)
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x) = x/(1-x^2). [Joerg Arndt, Aug 06 2012]
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MATHEMATICA
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a1350[n_] := Sum[Binomial[k - 1, 2k - n] n/(n - k), {k, 0, n - 1}];
a[n_] := 1/n Sum[EulerPhi[n/d] a1350[d], {d, Divisors[n]}];
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PROG
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(PARI)
N=66; x='x+O('x^N);
B(x)=x/(1-x^2);
A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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