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A006300 Number of rooted maps with n edges on torus.
(Formerly M5097)
1, 20, 307, 4280, 56914, 736568, 9370183, 117822512, 1469283166, 18210135416, 224636864830, 2760899996816, 33833099832484, 413610917006000, 5046403030066927, 61468359153954656, 747672504476150374, 9083423595292949240, 110239596847544663002, 1336700736225591436496, 16195256987701502444284 (list; graph; refs; listen; history; text; internal format)



E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.

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

T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.


T. D. Noe, Table of n, a(n) for n=2..100

D. Arquès, Relations fonctionnelles et dénombrement des cartes pointées sur le tore, J. Combin. Theory Ser. B, 43 (1987), 253-274.

E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271.

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014)

A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).

A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, Discrete Mathematics, Volume 310, Issue 3, 6 February 2010, Pages 518-526.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

T. R. S. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol.502, pp.4-15, (September-2013).


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

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

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


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)


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 *)



A005159_ser(N) = my(x='x+O('x^(N+1))); (1 - sqrt(1-12*x))/(6*x);

A006300_ser(N) = my(y = A005159_ser(N+1)); y*(y-1)^2/(3*(y-2)^2*(y+2));

Vec(A006300_ser(21)) \\ Gheorghe Coserea, Jun 02 2017


Cf. A007137, A006386.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, this sequence, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

Sequence in context: A001755 A016190 A016188 * A282372 A240799 A281931

Adjacent sequences:  A006297 A006298 A006299 * A006301 A006302 A006303




N. J. A. Sloane


Bender et al. give 20 terms.

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005

More terms from Joerg Arndt, Feb 26 2014



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Last modified August 7 08:55 EDT 2020. Contains 336274 sequences. (Running on oeis4.)