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

Offset

2,2

References

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.

Links

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).

S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

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, pp. 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).

Formula

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

Maple

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)

Mathematica

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 *)

Prog

(PARI)
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

Crossrefs

Column k=1 of A238396.

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: A361577 A016190 A016188 * A282372 A361293 A240799

Adjacent sequences: A006297 A006298 A006299 * A006301 A006302 A006303

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved