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%I M5097 #51 Oct 14 2023 23:53:54
%S 1,20,307,4280,56914,736568,9370183,117822512,1469283166,18210135416,
%T 224636864830,2760899996816,33833099832484,413610917006000,
%U 5046403030066927,61468359153954656,747672504476150374,9083423595292949240,110239596847544663002,1336700736225591436496,16195256987701502444284
%N Number of rooted maps with n edges on torus.
%D E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.
%H T. D. Noe, <a href="/A006300/b006300.txt">Table of n, a(n) for n = 2..100</a>
%H D. Arquès, <a href="http://dx.doi.org/10.1016/0095-8956(87)90002-5">Relations fonctionnelles et dénombrement des cartes pointées sur le tore</a>, J. Combin. Theory Ser. B, 43 (1987), 253-274.
%H E. A. Bender, E. R. Canfield and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CMB-1988-039-4">The enumeration of maps on the torus and the projective plane</a>, Canad. Math. Bull., 31 (1988), 257-271.
%H Sean R. Carrell, Guillaume Chapuy, <a href="http://arxiv.org/abs/1402.6300">Simple recurrence formulas to count maps on orientable surfaces</a>, arXiv:1402.6300 [math.CO], (19-March-2014).
%H A. D. Mednykh and R. Nedela, <a href="http://www.savbb.sk/mu/articles/4_2004_nedela.pdf">Enumeration of unrooted maps with given genus</a>, preprint (submitted to J. Combin. Th. B).
%H A. D. Mednykh and R. Nedela, <a href="http://dx.doi.org/10.1016/j.disc.2009.03.033">Enumeration of unrooted maps with given genus</a>, Discrete Mathematics, Volume 310, Issue 3, 6 February 2010, pp. 518-526.
%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(72)90056-1">Counting rooted maps by genus</a>, J. Comb. Thy B13 (1972), 122-141 and 192-218.
%H T. R. S. Walsh, <a href="http://www.sciencedirect.com/science/article/pii/S0304397511007146">Counting maps on doughnuts</a>, Theoretical Computer Science, vol. 502, pp. 4-15, (September-2013).
%F G.f.: (R-1)^2/(12*R^2*(R+2)) where R=sqrt(1-12*x); a(n) is asymptotic to 12^n/24. - Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
%F a(n) = Sum_{k=0..n-2} 2^(n-3-k)*(3^(n-1)-3^k)*binomial(n+k,k). - _Ruperto Corso_, Dec 18 2011
%F D-finite with recurrence: n*a(n) +22*(-n+1)*a(n-1) +4*(22*n-65)*a(n-2) +96*(5*n-4)*a(n-3) +576*(-2*n+7)*a(n-4)=0. - _R. J. Mathar_, Feb 20 2020
%p R:=sqrt(1-12*x): seq(coeff(convert(series((R-1)^2/(12*R^2*(R+2)),x,50),polynom),x,n),n=2..25); (Pab Ter)
%t Drop[With[{c=Sqrt[1-12x]},CoefficientList[Series[(c-1)^2/(12c^2 (c+2)), {x,0,30}],x]],2] (* _Harvey P. Dale_, Jun 14 2011 *)
%o (PARI)
%o A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);
%o A006300_ser(N) = my(y = A005159_ser(N+1)); y*(y-1)^2/(3*(y-2)^2*(y+2));
%o Vec(A006300_ser(21)) \\ _Gheorghe Coserea_, Jun 02 2017
%Y Column k=1 of A238396.
%Y Cf. A007137, A006386.
%Y Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, this sequence, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
%K nonn,nice
%O 2,2
%A _N. J. A. Sloane_
%E Bender et al. give 20 terms.
%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005
%E More terms from _Joerg Arndt_, Feb 26 2014
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