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 A032189 Number of ways to partition n elements into pie slices each with an odd number of elements. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 LINKS C. G. Bower, Transforms (2) P. Flajolet and M. Soria, The Cycle Construction In SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60. Petros Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2. FORMULA a(n) = A000358(n)-(1+(-1)^n)/2. "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] a(n) = (1/n)*Sum_{d divides n} phi(n/d)*A001350(d). - Petros Hadjicostas, Dec 27 2016 MATHEMATICA 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]}]; Array[a, 50] (* Jean-François Alcover, Jul 29 2018, after Petros Hadjicostas *) PROG (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) /* Joerg Arndt, Aug 06 2012 */ CROSSREFS Cf. A000358, A001350, A008965. Sequence in context: A136422 A173674 A018128 * A316077 A186425 A034395 Adjacent sequences:  A032186 A032187 A032188 * A032190 A032191 A032192 KEYWORD nonn AUTHOR STATUS approved

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Last modified March 25 10:31 EDT 2019. Contains 321470 sequences. (Running on oeis4.)