A365857 Number of cyclic compositions of 2*n into odd parts.

1, 2, 4, 7, 14, 30, 63, 142, 328, 765, 1810, 4340, 10461, 25414, 62074, 152287, 375166, 927554, 2300347, 5721044, 14264308, 35646311, 89264834, 223959710, 562878429, 1416953362, 3572233420, 9018211989, 22795835726, 57690911720, 146164582455, 370705552702, 941109975022, 2391391374017, 6081865318124

Offset

1,2

Comments

Even bisection of A032189.

Also the number of cyclic compositions into an even number of odd parts; because such a sum must be even, alternating terms are zero and have been removed.

Also the number of dual classes of cyclic n-color compositions of n. A cyclic composition is a sum of positive integers in which the order of the parts is considered up to cyclic permutation. In other words, it is the collection of components remaining in the cycle graph C_n on n vertices when one or more edges are removed, and rotations are considered equivalent. In an n-color composition, each part of size k is assigned one of k "colors" which may be represented graphically by marking one vertex in the part. (See A032198 for the number of cyclic n-color compositions.) The dual of a cyclic n-color composition is obtained by switching the roles of edges and vertices in C_n, then removing each edge that came from a previously marked vertex while marking each vertex that came from a previously removed edge. Each cyclic n-color composition of n either belongs to a dual pair or is self-dual. (See A365859 for the number of self-dual cyclic n-color compositions.)

Links

Table of n, a(n) for n=1..35.

A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.

Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.

Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color cyclic compositions, Discrete Mathematics 341 (2018), 3209-3226.

Formula

G.f.: (1/2)*(Sum_{k>=1} phi(k)/k * log((1-2*x^k+x^(2*k))/(1-3*x^k+x^(2*k))) + Sum_{m>=1} phi(2*m)/(2*m) * log((1+x^m-x^(2*m))/(1-x^m-x^(2*m)))).

a(n) = (1/(2*n)) * Sum_{k divides 2*n} phi(k)*A001350((2*n)/k).

a(n) = (A032198(n) + A365859(n))/2.

Prog

(PARI)
N=99;  x='x+O('x^N); B(x)=x/(1-x^2);
A=Vec(sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k)))));
vector(#A\2, n, A[2*n]) \\ Joerg Arndt, Sep 22 2023
(Python)
from sympy import totient, lucas, divisors
def A365857(n): return sum(totient((n<<1)//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n<<1, generator=True))//n>>1 # Chai Wah Wu, Sep 23 2023

Crossrefs

Cf. A001350, A032189, A032198, A365858, A365859.

Sequence in context: A157133 A367462 A202850 * A247295 A120262 A202849

Adjacent sequences: A365854 A365855 A365856 * A365858 A365859 A365860

Keyword

nonn

Author

Joshua P. Bowman, Sep 20 2023

Status

approved