

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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



FORMULA

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.
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&;


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



KEYWORD

nonn


AUTHOR



STATUS

approved



