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%I M1260 N0482 #110 Nov 23 2022 08:57:10
%S 1,2,4,12,39,202,1219,9468,83435,836017,9223092,111255228,1453132944,
%T 20433309147,307690667072,4940118795869,84241805734539,
%U 1520564059349452,28963120073957838,580578894859915650,12217399235411398127,269291841184184374868,6204484017822892034404
%N Number of n-gons with n vertices.
%C Number of inequivalent undirected Hamiltonian cycles in complete graph on n labeled nodes under action of dihedral group of order 2n acting on nodes.
%D J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%D R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ludovic Schwob, <a href="/A000940/b000940.txt">Table of n, a(n) for n = 3..500</a> (terms 3..100 from T. D. Noe).
%H S. W. Golomb and L. R. Welch, <a href="http://www.jstor.org/stable/2308978">On the enumeration of polygons</a>, Amer. Math. Monthly, 67 (1960), 349-353.
%H S. W. Golomb and L. R. Welch, <a href="/A000939/a000939.pdf">On the enumeration of polygons</a>, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]
%H Samuel Herman and Eirini Poimenidou, <a href="https://arxiv.org/abs/1905.04785">Orbits of Hamiltonian Paths and Cycles in Complete Graphs</a>, arXiv:1905.04785 [math.CO], 2019.
%H E. M. Palmer and R. W. Robinson, <a href="http://dx.doi.org/10.1007/BF02392038">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%H R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a> (gives initial terms of this sequence, except he has a(6)=7 instead of 12)
%H R. C. Read, <a href="/A002832/a002832.pdf">Letter to N. J. A. Sloane, 1992</a>
%H R. C. Read, <a href="http://dx.doi.org/10.1016/S0012-365X(96)00255-5">Combinatorial problems in theory of music</a>, Discrete Math. 167 (1997), 543-551.
%H N. J. A. Sloane, <a href="/A000940/a000940.jpg">Illustration of initial terms</a> [Annotated page from Golomb-Welch article]
%H Venta Terauds and J. Sumner, <a href="https://arxiv.org/abs/1712.00858">Circular genome rearrangement models: applying representation theory to evolutionary distance calculations</a>, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.
%H Venta Terauds and Jeremy Sumner, <a href="https://arxiv.org/abs/2012.11665">A new algebraic approach to genome rearrangement models</a>, arXiv:2012.11665 [q-bio.PE], 2020.
%F For formula see Maple lines.
%F a(p) = ((((p-1)! + 1)/p) + p - 2 + (2^((p-1)/2)*((p-1)/2)!))/4 for prime p. See A007619. - _Ian Mooney_, Oct 05 2022
%F a(n) ~ sqrt(2*Pi)/4 * n^(n-3/2) / e^n. - _Ludovic Schwob_, Nov 03 2022
%e Label the vertices of a regular n-gon 1,2,...,n.
%e For n=3,4,5 representatives for the polygons counted here are:
%e (1,2,3,1),
%e (1,2,3,4,1), (1,2,4,3,1),
%e (1,2,3,4,5,1), (1,2,3,5,4,1), (1,2,4,5,3,1), (1,3,5,2,4,1).
%e For n=6:
%e (1,2,3,4,5,6,1), (1,2,3,4,6,5,1), (1,2,3,5,6,4,1),
%e (1,2,3,6,5,4,1), (1,2,4,3,6,5,1), (1,2,4,6,3,5,1),
%e (1,2,4,6,5,3,1), (1,2,5,3,6,4,1), (1,2,5,4,6,3,1),
%e (1,2,5,6,3,4,1), (1,2,6,4,5,3,1), (1,3,5,2,6,4,1).
%p with(numtheory);
%p # for n odd:
%p Sd:=proc(n) local t1,d; t1:=2^((n-1)/2)*n^2*((n-1)/2)!; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
%p # for n even:
%p Se:=proc(n) local t1,d; t1:=2^(n/2)*n*(n+6)*(n/2)!/4; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
%p A000940:=n-> if n mod 2 = 0 then Se(n) else Sd(n); fi;
%t a[n_] := (t1 = If[OddQ[n], 2^((n - 1)/2)*n^2*((n - 1)/2)!, 2^(n/2)*n*(n + 6)*(n/2)!/4]; For[ d = 1 , d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[n/d]^2*d!*(n/d)^d]]; t1/(4*n^2)); Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, Jun 19 2012, after Maple *)
%o (PARI) a(n)={if(n<3, 0, (2^(n\2-2)*(n\2)!*n*if(n%2, 4*n, n + 6) + sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d))/(4*n^2))} \\ _Andrew Howroyd_, Sep 09 2018
%o (Python)
%o from sympy import factorial, divisors, totient
%o def A000940(n): return 1 if n == 3 else ((sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True))+(1<<(k:=n>>1)-2)*n*(n<<2 if n&1 else (n+6))*factorial(k))>>2)//n//n # _Chai Wah Wu_, Nov 07 2022
%Y Cf. A000939, A007619. Bisections give A094156, A094157.
%Y For permutation classes under various symmetries see A089066, A262480, A002619.
%K nonn,easy,nice
%O 3,2
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004
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