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A032287
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"DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...
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5
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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
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1,2
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COMMENTS
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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)
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LINKS
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FORMULA
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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.
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)
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MATHEMATICA
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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&;
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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