

A005648


Number of 2nbead blackwhite reversible necklaces with n black beads.
(Formerly M0878)


13



1, 1, 2, 3, 8, 16, 50, 133, 440, 1387, 4752, 16159, 56822, 200474, 718146, 2587018, 9398520, 34324174, 126068558, 465093571, 1723176308, 6407924300, 23910576230, 89494164973, 335913918902, 1264107416466
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OFFSET

0,3


COMMENTS

a(n) is the coefficient of c_1^n*c_2^n in the cycle index polynomial for the dihedral group D_{2*n} evaluated with the figure counting polynomial c = c_1 + c_2, n>=1, abbreviated as Z(D_{2*n},c). See, e.g., the HararyPalmer reference (given under A212355), p. 42, Theorem (PET), and the example for all 6 twocolored 4bracelets (called there necklaces) on p. 44, Figure 2.4.2.  Wolfdieter Lang, Jun 05 2012


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

a(n) = ( Sum_{dn} phi(n/d)*C(2*d, d) )/(4*n) + C(2*k, k)/2, where k = floor(n/2).  Michael Somos
a(n) = (A003239(n) + C(2*k, k))/2, where k = [ n/2 ].  R. J. Fletcher, (yylee(AT)mail.ncku.edu.tw)


EXAMPLE

a(2) = 2: BBWW, BWBW.
a(3) = 3: BBBWWW, BBWBWW, BWBWBW.
a(4) = 8: BBBBWWWW, BBBWBWWW, BBBWWBWW, BBWWBBWW, BBWBWBWW, BBWBWWBW, BBWBBWWW, BWBWBWBW.


MATHEMATICA

f[k_Integer, n_] := (Plus @@ (EulerPhi[ # ]Binomial[n/#, k/# ] & /@ Divisors[GCD[n, k]])/n + Binomial[(n  If[OddQ@n, 1, If[OddQ@k, 2, 0]])/2, (k  If[OddQ@k, 1, 0])/2])/2 (* Robert A. Russell, Sep 27 2004 *)
a[0] = 1; a[n_] := 1/2*(Binomial[2*Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#]*Binomial[2*n/#, n/#]&]/(2*n)); Array[a, 26, 0] (* JeanFrançois Alcover, Nov 05 2017, translated from PARI *)


PROG

(PARI) a(n) = 1/2*( binomial(2*(n\2), n\2) + if(n<1, n >= 0, sumdiv(n, k, eulerphi(k)*binomial(2*n/k, n/k))/(2*n) ));


CROSSREFS



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



