cp's OEIS Frontend

A112849 Number of congruence classes (epimorphisms/vertex partitionings induced by graph endomorphisms) of undirected cycles of even length: |C(C_{2*n})|.

%I A112849 #34 Sep 08 2022 08:45:23
%S A112849 1,4,11,36,127,463,1717,6436,24311,92379,352717,1352079,5200301,
%T A112849 20058301,77558761,300540196,1166803111,4537567651,17672631901,
%U A112849 68923264411,269128937221,1052049481861,4116715363801,16123801841551,63205303218877,247959266474053,973469712824057,3824345300380221,15033633249770521
%N A112849 Number of congruence classes (epimorphisms/vertex partitionings induced by graph endomorphisms) of undirected cycles of even length: |C(C_{2*n})|.
%D A112849 M. A. Michels, About The Structure of Graph Endomorphisms, Diploma thesis, University of Oldenburg, Germany, 2005.
%H A112849 Vincenzo Librandi, <a href="/A112849/b112849.txt">Table of n, a(n) for n = 1..1000</a>
%H A112849 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A112849 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]
%H A112849 M. A. Michels and U. Knauer, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.022">The congruence classes of paths and cycles</a>, Discrete Math., 309 (2009), 5352-5359. [_N. J. A. Sloane_, Sep 15 2009]
%F A112849 |C(C_2n)| = 1 + (1/2)*binomial(2*n-1, n-1) + (1/2)*binomial(2*n-1, n), n > 1.
%F A112849 a(n) = A260878(n) for n >= 2. - _Alois P. Heinz_, Aug 06 2015
%F A112849 Conjecture: n*(3*n - 5)*a(n) + (-15*n^2 + 31*n - 12)*a(n-1) + 2*(3*n - 2)*(2*n - 3)*a(n-2) = 0. - _R. J. Mathar_, Aug 07 2015
%p A112849 egf := n->exp(exp(x)*(1-(GAMMA(n,x)/GAMMA(n)))):
%p A112849 a := n->`if`(n=1,1,(2*n)!*coeff(series(egf(n),x,2*n+1),x,2*n)):
%p A112849 seq(a(n),n=1..29); # _Peter Luschny_, Apr 05 2011
%t A112849 Join[{1}, Table[1 + (1/2) Binomial[2 n - 1, n - 1] + (1/2)Binomial[2 n - 1, n], {n, 2, 30}]] (* _Vincenzo Librandi_, Feb 26 2017 *)
%o A112849 (Magma) [1] cat [1 + (1/2)*Binomial(2*n-1, n-1) + (1/2)*Binomial(2*n-1, n): n in [2..30]]; // _Vincenzo Librandi_, Feb 26 2017
%o A112849 (PARI) a(n) = if (n==1, 1, 1 + (binomial(2*n-1, n-1) + binomial(2*n-1, n))/2); \\ _Michel Marcus_, Feb 26 2017
%Y A112849 Cf. A112850, A260878.
%K A112849 easy,nonn
%O A112849 1,2
%A A112849 Martin Alexander Michels (martinmichels(AT)t-online.de), Sep 24 2005