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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

Table of n, a(n) for n=1..47.

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.

P. Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.

Index entries for sequences related to necklaces

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

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 * A186425 A034395 A032232

Adjacent sequences:  A032186 A032187 A032188 * A032190 A032191 A032192

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified January 16 09:46 EST 2017. Contains 280814 sequences.