A007137 -id:A007137 - cp's OEIS frontend

A006300 Number of rooted maps with n edges on torus.

1, 20, 307, 4280, 56914, 736568, 9370183, 117822512, 1469283166, 18210135416, 224636864830, 2760899996816, 33833099832484, 413610917006000, 5046403030066927, 61468359153954656, 747672504476150374, 9083423595292949240, 110239596847544663002, 1336700736225591436496, 16195256987701502444284

Offset: 2

N. J. A. Sloane

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).
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).

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.

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

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

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

A267180 Triangle read by rows: T(n,k) = number of rooted maps with n edges on a nonorientable surface of genus k (1 <= k <= n).

Original entry on oeis.org

1, 10, 4, 98, 84, 41, 982, 1340, 1380, 488, 10062, 19280, 31225, 23320, 8229, 105024, 263284, 592824, 696912, 516958, 164892, 1112757, 3486224, 10185056, 16662492, 19381145, 12980716, 4016613, 11934910, 45247084, 164037704, 348539072, 562395292, 590136856, 382630152, 112818960

Offset: 1

N. J. A. Sloane, Jan 16 2016

			Triangle begins:
1,
10,4,
98,84,41,
982,1340,1380,488,
10062,19280,31225,23320,8229,
105024,263284,592824,696912,516958,164892,
1112757,3486224,10185056,16662492,19381145,12980716,4016613,
11934910,45247084,164037704,348539072,562395292,590136856,382630152,112818960
...

David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

See A238396 for analog for orientable surfaces.
Columns give A007137, A006344.
Cf. A380235 (unrooted).

A316598 a(n) is the number of rooted quadrangulations of the projective plane with n vertices.

Original entry on oeis.org

5, 38, 331, 3098, 30330, 306276, 3163737, 33252050, 354312946, 3817498004, 41510761346, 454882507468, 5017662052868, 55664182358808, 620592559670979, 6949200032479746, 78117065527443654, 881170275583541004, 9970663315885385502, 113137928354523348300

Offset: 1

Gheorghe Coserea, Jul 08 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..301
Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008-2009.
E. Krasko, A. Omelchenko, Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps, arXiv:1709.03225 [math.CO], 2017.
E. Krasko, A. Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics, Volume 342, Issue 2, February 2019, Pages 584-599.

Cf. A005159, A007137, A278120.

(-6*x + 3*Sqrt[1-12*x] - 2*Sqrt[-36*x + 6*Sqrt[1-12*x] + 3] + 3)/(6*x^2) + O[x]^20 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 24 2019 *)

seq(N) = {
  my(x='x + O('x^(N+2)), c=(1-sqrt(1-12*x))/(6*x));
  Vec((1 - x - 3*x*c - sqrt(1 - 4*x - 4*x*c))/x);
};
seq(20)
\\ test: y='x*Ser(seq(300), 'x); 0 == 3*x^3*y^4 + (12*x^3 - 6*x^2)*y^3 + (18*x^3 + 24*x^2 + x)*y^2 + (12*x^3 + 66*x^2 + 8*x - 2)*y + (3*x^3 + 36*x^2 + 10*x)

G.f. A(x) = (1-x-3*x*c - sqrt(1-4*x-4*x*c))/x, where c=(1-sqrt(1-12*x))/(6*x). (see eqn. (117) in Garoufalidis link)
G.f. y=A(x) satisfies:
0 = 3*x^3*y^4 + 6*x^2*(2*x - 1)*y^3 + x*(18*x^2 + 24*x + 1)*y^2 + 2*(6*x^3 + 33*x^2 + 4*x - 1)*y + x*(3*x^2 + 36*x + 10).
0 = 13*x*(4*x + 1)*(12*x - 1)^3*y''''' + (36864*x^4 + 3840*x^3 + 8832*x^2 + 1556*x - 65)*(12*x - 1)^2*y'''' + 16*(248832*x^4 - 5184*x^3 + 29799*x^2 + 2418*x - 259)*(12*x - 1)*y''' + 72*(1382400*x^4 - 201600*x^3 + 144312*x^2 - 4157*x - 492)*y'' + 144*(276480*x^3 - 51840*x^2 + 31488*x - 979)*y' + 165888*y.
0 = x*(4*x + 1)*(48*x^2 - 6*x + 1)*(12*x - 1)^3*y'''' + 2*(10368*x^4 + 12*x^2 + 47*x - 2)*(12*x - 1)^2*y''' + 6*(86400*x^4 - 10800*x^3 + 2472*x^2 + 132*x - 19)*(12*x - 1)*y'' + (2488320*x^4 - 622080*x^3 + 186192*x^2 - 10728*x - 144)*y' + (10368*x - 648)*y.
a(n) ~ 2^(2*n + 1/2) * 3^(n + 1/2)/ (Gamma(3/4) * n^(5/4)) * (1 - sqrt(3) * Gamma(3/4) / (sqrt(2*Pi) * n^(1/4))). - Vaclav Kotesovec, Oct 04 2019

A316698 a(n) is the number of rooted 2-connected triangular maps on the projective plane with n vertices.

Original entry on oeis.org

0, 0, 1, 18, 261, 3539, 46695, 608526, 7884661, 101905839, 1316047599, 16998339587, 219699143367, 2842235616645, 36809980380883, 477280717428102, 6195737611180053, 80522713890559319, 1047702563499718623, 13646946767000964471, 177947654115176898479

Offset: 1

Gheorghe Coserea, Jul 10 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..303
Zhi-Cheng Gao, The number of rooted 2-connected triangular maps on the projective plane, Journal of Combinatorial Theory, Series B, Volume 53, Issue 1, September 1991, Pages 130-142.

Cf. A007137, A316598.

seq(N) = {
  my(x = 'x + O('x^(N+1)), r=serreverse(x*(1-2*x)^2),
     v = Vec(subst((1-sqrt((1-6*x)/(1-2*x)))/(2*x)-1/(1-3*x), 'x, r)));
  concat([0,0], v);
};
seq(21)

G.f. A(x) = (1 - sqrt((1-6*r)/(1-2*r)))/(2*r) - 1/(1-3*r), where r(x) satisfies x = r*(1-2*r)^2, with r(0)=0. (see (1.1) in Gao link)
G.f. y=A(x) satisfies: 0 = (729*x^2 - 54*x + 1)*y^6 + (-567*x^2 + 48*x - 1)*y^5 + (4617*x^3 - 486*x^2 + 12*x)*y^4 + (-14310*x^4 + 1772*x^3 - 54*x^2)*y^3 + (-672*x^4 + 50*x^3)*y^2 + (126*x^5 - 36*x^4 + 2*x^3)*y - 2*x^6.
Recurrence: (n-1)*n*(2*n - 3)*(4*n - 9)*(4*n - 3)*(972*n^7 - 1944*n^6 - 169443*n^5 + 1865607*n^4 - 8817457*n^3 + 21764795*n^2 - 27508222*n + 14065464)*a(n) = 3*(n-1)*(699840*n^11 - 5598720*n^10 - 107581284*n^9 + 2120974416*n^8 - 16716827583*n^7 + 77044659801*n^6 - 229110154570*n^5 + 453176543549*n^4 - 592757452327*n^3 + 491840891214*n^2 - 233773288056*n + 48250762560)*a(n-1) - 9*(5668704*n^12 - 61410960*n^11 - 770480100*n^10 + 20379495348*n^9 - 192680893665*n^8 + 1066797111051*n^7 - 3886131103119*n^6 + 9712411159089*n^5 - 16796662782944*n^4 + 19765806847064*n^3 - 15086450010036*n^2 + 6716653116768*n - 1318624045200)*a(n-2) + 486*(3*n - 11)*(3*n - 10)*(122472*n^10 - 717336*n^9 - 21548106*n^8 + 353617272*n^7 - 2470176720*n^6 + 10020300957*n^5 - 25599297354*n^4 + 41773597853*n^3 - 42167708852*n^2 + 23887121874*n - 5766718860)*a(n-3) - 26244*(n-4)*(3*n - 14)*(3*n - 13)*(3*n - 11)*(3*n - 10)*(972*n^7 + 4860*n^6 - 160695*n^5 + 1023252*n^4 - 3054319*n^3 + 4802888*n^2 - 3820650*n + 1199772)*a(n-4). - Vaclav Kotesovec, Jul 11 2018
a(n) ~ (27/2)^n * (1/(2*3^(7/4)*Gamma(3/4)) - 10/(27*sqrt(3*Pi)*n^(1/4)) + sqrt(2)*Gamma(3/4) / (3^(9/4)*Pi*sqrt(n))) / n^(5/4) [main asymptotic term by Gao, 1991]. - Vaclav Kotesovec, Jul 11 2018

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A006300 Number of rooted maps with n edges on torus.

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A267180 Triangle read by rows: T(n,k) = number of rooted maps with n edges on a nonorientable surface of genus k (1 <= k <= n).

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A316598 a(n) is the number of rooted quadrangulations of the projective plane with n vertices.

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A316698 a(n) is the number of rooted 2-connected triangular maps on the projective plane with n vertices.

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