

A032287


"DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...


5



1, 3, 6, 13, 24, 51, 97, 207, 428, 946, 2088, 4831, 11209, 26717, 64058, 155725, 380400, 936575, 2314105, 5744700, 14300416, 35708268, 89359536, 224121973, 563126689, 1417378191, 3572884062, 9019324297, 22797540648
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OFFSET

1,2


COMMENTS

From Petros Hadjicostas, Jun 21 2019: (Start)
Under Bower's transforms, the input sequence c = (c(m): m >= 1) describes how each part of size m in a composition is colored. In a composition (ordered partition) of n >= 1, a part of size m is assumed to be colored with one of c(m) colors.
Under the DIK transform, we are dealing with "dihedral compositions" of n >= 1. These are equivalence classes of ordered partitions of n such that two such ordered partitions are equivalent if one can be obtained from the other by rotation or reflection.
If the input sequence is c = (c(m): m >= 1), denote the output sequence under the DIK transform by b = (b(n): n >= 1); i.e., b(n) = (DIK c)(n) for n >= 1. If C(x) = Sum_{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the g.f. of b = DIK c is Sum_{n >= 1} b(n)*x^n = (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1  C(x^d)) + (1 + C(x))^2/(4 * (1  C(x^2)))  (1/4).
For the current sequence (a(n): n >= 1), the input sequence is c(m) = m for all m >= 1. That is, we are dealing with the socalled "mcolor dihedral compositions". Here, a(n) is the number of dihedral compositions of n where each part of size m may be colored with one of m colors. For the linear and cyclic versions of such mcolor compositions, see Agarwal (2000), Gibson (2017), and Gibson et al. (2018).
Since C(x) = x/(1  x)^2, we have Sum_{n >= 1} a(n) * x^n = (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1  x^d)^2 / (1  3*x^d + x^(2*d))) + (1/2) * x * (1 + x  2*x^2 + x^3 + x^4)/((1  x)^2 * (1 + x  x^2) * (1  x  x^2)), which is the g.f. given by Andrew Howroyd in the PARI program below.
Note that Sum_{d >= 1} (phi(d)/d) * log (1  C(x^d)) = Sum_{d >= 1} (phi(d)/d) * log((1  x^d)^2 / (1  3*x^d + x^(2*d))) is the g.f. of the "mcolor cyclic compositions" that appear in Gibson (2017) and Gibson et al. (2018). See sequence A032198, which is the CIK transform of sequence (c(m): m >= 1) = (m: m >= 1).
(End)


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200
A. K. Agarwal, ncolour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 14211427.
C. G. Bower, Transforms (2).
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of ncolor compositions, Discrete Mathematics 341 (2018), 32093226.
Arnold Knopfmacher and Neville Robbins, Some properties of dihedral compositions, Util. Math. 92 (2013), 207220.
Index entries for sequences related to bracelets


FORMULA

From Petros Hadjicostas, Jun 21 2019: (Start)
a(n) = ( F(n+4) + (1)^n * F(n4)  2 * (n + 1) + (1/n) * Sum_{dn} phi(n/d) * L(2*d) )/2 for n >= 4, where F(n) = A000045(n) and L(n) = A000032(n) are the usual nth Fibonacci and nth Lucas numbers, respectively.
a(n) = (A032198(n) + A308747(n))/2 for n >= 1.
G.f.: (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1  x^d)^2 / (1  3*x^d + x^(2*d))) + (1/2) * x * (1 + x  2*x^2 + x^3 + x^4)/((1  x)^2 * (1 + x  x^2) * (1  x  x^2)).
(End)


MATHEMATICA

seq[n_] := x(1 + x  2 x^2 + x^3 + x^4)/((1  x)^2 (1  x  x^2)(1 + x  x^2)) + Sum[EulerPhi[d]/d Log[(1  x^d)^2/(1  3 x^d + x^(2d)) + O[x]^(n+1)], {d, 1, n}] // CoefficientList[#, x]& // Rest // #/2&;
seq[30] (* JeanFrançois Alcover, Sep 17 2019, from PARI *)


PROG

(PARI) seq(n)={Vec(x*(1 + x  2*x^2 + x^3 + x^4)/((1  x)^2*(1  x  x^2)*(1 + x  x^2)) + sum(d=1, n, eulerphi(d)/d*log((1  x^d)^2/(1  3*x^d + x^(2*d)) + O(x*x^n))))/2} \\ Andrew Howroyd, Jun 20 2018


CROSSREFS

Cf. A000032, A000045, A001906, A005594, A032198, A088305, A308747.
Sequence in context: A027999 A005196 A320286 * A199403 A006017 A147323
Adjacent sequences: A032284 A032285 A032286 * A032288 A032289 A032290


KEYWORD

nonn


AUTHOR

Christian G. Bower


STATUS

approved



