A032287 "DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...

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

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

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 so-called "m-color 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 m-color 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 "m-color 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, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
• 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 n-color compositions, Discrete Mathematics 341 (2018), 3209-3226.
• Arnold Knopfmacher and Neville Robbins, Some properties of dihedral compositions, Util. Math. 92 (2013), 207-220.
• Index entries for sequences related to bracelets

Crossrefs

Cf. A000032, A000045, A001906, A005594, A032198, A088305, A308747.

Programs

Maple

DIK := proc(L::list)
    local  x,cidx,ncyc,d,gd,g,g2,n ;
    n := nops(L) ;
    g := add(op(i,L)*x^i,i=1..n) ;
    # wrap into the cycle index of the cyclic group C_n
    cidx := 0 ;
    for ncyc from 1 to n do
        for d in numtheory[divisors](ncyc) do
            gd := subs(x=x^d,g) ;
            cidx := cidx+1/ncyc*numtheory[phi](d)*gd^(ncyc/d) ;
        end do:
    end do:
    # cycle index is half of th eone for the cyclic group plus two
    # different branches or D_n with even or odd n
    cidx := cidx/2 ;
    g2 := subs(x=x^2,g) ;
    for ncyc from 1 to n do
        if type(ncyc,'odd') then
            cidx := cidx+ g*g2^((ncyc-1)/2)/2 ;
        else
            cidx := cidx+ (g^2*g2^((ncyc-2)/2) + g2^(ncyc/2))/4 ;
        end if;
    end do:
    taylor(cidx,x=0,nops(L)) ;
    gfun[seriestolist](%) ;
end proc:
A032287_list := proc(n)
        local ele ;
        ele := [seq(i,i=1..40)] ;
        DIK(ele) ;
end proc:
A032287_list(50) ; # R. J. Mathar, Feb 14 2025

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] (* Jean-François Alcover, Sep 17 2019, from PARI *)

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

Formula

From Petros Hadjicostas, Jun 21 2019: (Start)

a(n) = ( F(n+4) + (-1)^n * F(n-4) - 2 * (n + 1) + (1/n) * Sum_{d|n} phi(n/d) * L(2*d) )/2 for n >= 4, where F(n) = A000045(n) and L(n) = A000032(n) are the usual n-th Fibonacci and n-th 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)