A006300 -id:A006300 - cp's OEIS frontend

A238359 Number of genus-9 rooted maps with n edges.

11665426077721040625, 3498878057690404966500, 540996834819906946713375, 57494374008560749302297480, 4724172886681078698955547790, 320061005837218582787265273000, 18618409220753939214291224549409, 956146512935178711199035220384800, 44232688287025023758415781081779828, 1871678026675570344184400604255444240

Offset: 18

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 18..200
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Gheorghe Coserea, The g.f. as a rational function of y=A005159(x)
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Column g=9 of A269919.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, this sequence, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 9];
Table[a[n], {n, 18, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A238360 Number of genus-10 rooted maps with n edges.

Original entry on oeis.org

15230046989184655753125, 5199629454143883380553750, 909887917857275652461097750, 108861830345440643086946970900, 10021124647635764856828690342402, 757187906770815991999545249667404, 48918614114003431712044170834572688, 2779227352989564224315657269511192976, 141720718575991991799057452443438430580

Offset: 20

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 20..200
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Gheorghe Coserea, The g.f. as a rational function of y=A005159(x)
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Column g=10 of A269919.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, this sequence.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 10];
Table[a[n], {n, 20, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A006386 Number of sensed genus 1 maps with n edges.

Original entry on oeis.org

1, 6, 46, 452, 4852, 52972, 587047, 6550808, 73483256, 827801468, 9360123740, 106189359544, 1208328304864, 13787042250528, 157700137398689, 1807893066408464, 20768681225892328, 239037464947999900, 2755989928117365244, 31826208029615881656, 368074022535205870382

Offset: 2

N. J. A. Sloane

A genus 1 map can be called a toroidal map.

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

T. D. Noe, Table of n, a(n) for n = 2..30 (from Mednykh and Nedela)
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, J. Combin. Th. B, 96 (2006), 706-729.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol.502, pp.4-15, (September-2013).
Timothy R. S. Walsh, Alain Giorgetti, Alexander Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), no. 17, 2660--2671. MR2935417. - From _N. J. A. Sloane_, Aug 01 2012

Column k=1 of A379438.

Cf. A006300 (rooted), A006384 (planar), A006387 (unsensed), A104595, A104596, A215019.

More terms from Valery A. Liskovets, Mar 22 2005

Edited by N. J. A. Sloane, May 23 2008

A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 0, 9, 1, 0, 54, 20, 0, 0, 378, 307, 21, 0, 0, 2916, 4280, 966, 0, 0, 0, 24057, 56914, 27954, 1485, 0, 0, 0, 208494, 736568, 650076, 113256, 0, 0, 0, 0, 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0, 17399772, 117822512, 248371380, 167808024, 24635754, 0, 0, 0, 0, 0, 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0

Offset: 0

Joerg Arndt, Feb 26 2014

			Triangle starts:
00: 1,
01: 2, 0,
02: 9, 1, 0,
03: 54, 20, 0, 0,
04: 378, 307, 21, 0, 0,
05: 2916, 4280, 966, 0, 0, 0,
06: 24057, 56914, 27954, 1485, 0, 0, 0,
07: 208494, 736568, 650076, 113256, 0, 0, 0, 0,
08: 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0,
09: 17399772, 117822512, 248371380, 167808024, 24635754, 0, ...,
10: 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0, ...,
11: 1602117468, 18210135416, 73231116024, 117593590752, 66519597474, 8608033980, 0, ...,
12: 15792300756, 224636864830, 1183803697278, 2675326679856, 2416610807964, 672868675017, 24325703325, 0, ...,
...

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.

Joerg Arndt, Table of n, a(n) for n = 0..1325 (rows 0..50, flattened)
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).

Columns k for 0<=k<=10 are: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Sum of row n is A000698(n+1).
See A267180 for nonorientable analog.
Cf. A269418, A269419.
The triangle without the zeros is A269919.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4n - 2)/3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n - k)-1) T[k-1, i] T[n-k-1, g-i] , {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Gheorghe Coserea *)

N=20;
MEM=matrix(N+1,N+1, r,c, -1);  \\ for memoization
Q(n,g)=
{
    if (n<0,  return( (g<=0) ) ); \\ not given in paper
    if (g<0,  return( 0 ) ); \\ not given in paper
    if (n<=0, return( g==0 ) );  \\ as in paper
    my( m = MEM[n+1,g+1] );
    if ( m != -1,  return(m) );  \\ memoized value
    my( t=0 );
    t += (4*n-2)/3 * Q(n-1, g);
    t += (2*n-3)*(2*n-2)*(2*n-1)/12 * Q(n-2, g-1);
    my(l, j);
    t += 1/2*
        sum(k=1, n-1, l=n-k;  \\ l+k == n, both >= 1
            sum(i=0, g, j=g-i;  \\ i+j == g, both >= 0
                (2*k-1)*(2*l-1) * Q(k-1, i) * Q(l-1, j)
            );
        );
    t *= 6/(n+1);
    MEM[n+1, g+1] = t;  \\ memoize
    return(t);
}
for (n=0, N, for (g=0, n, print1(Q(n, g),", "); );  print(); ); /* print triangle */

From Gheorghe Coserea, Mar 11 2016: (Start)
(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.
(End)

A006387 Number of unsensed genus 1 maps with n edges.

Original entry on oeis.org

0, 0, 1, 6, 40, 320, 2946, 29364, 309558, 3365108, 37246245, 416751008, 4696232371, 53186743416, 604690121555, 6896534910612, 78867385697513, 904046279771682, 10384916465797240, 119522063788612992, 1378014272286250059

Offset: 0

N. J. A. Sloane

nonn

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

Evgeniy Krasko and Alexander Omelchenko, Enumeration of Unsensed Orientable Maps on Surfaces of a Given Genus, arXiv:1712.10139 [math.CO], 2017.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Column k=1 of A379439.

Cf. A006300 (rooted), A006385 (planar), A006386 (sensed), A214814, A214815, A214816.

a(12)-a(20) from Evgeniy Krasko, Sep 17 2017

A007137 Number of rooted maps with n edges on the projective plane.

Original entry on oeis.org

1, 10, 98, 982, 10062, 105024, 1112757, 11934910, 129307100, 1412855500, 15548498902, 172168201088, 1916619748084, 21436209373224, 240741065193282, 2713584138389838, 30687358107371442, 348061628432108352

Offset: 1

N. J. A. Sloane

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
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; see p. 270.
Guillaume Chapuy, Maciej Dołęga, A bijection for rooted maps on general surfaces, arXiv:1501.06942 [math.CO], 2016; see corollary 4.5.
Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087). - _N. J. A. Sloane_, Jun 03 2012

Cf. A006300.

A column of A267180.

R:=sqrt(1-12*x): seq(coeff(convert(series(((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x),x,50),polynom),x,n),n=1..25); # Pab Ter, Nov 07 2005

With[{r=Sqrt[1-12x]},Rest[CoefficientList[Series[((2r+1)/3-Sqrt[r (r+2)/3])/ (2x),{x,0,20}],x]]](* Harvey P. Dale, Mar 02 2018 *)

seq(N) = {
  my(x = 'x + O('x^(N+2)), r=sqrt(1-12*x));
  Vec(((2*r+1)/3 - sqrt(r*(r+2)/3))/(2*x));
};
seq(18)
\\ test: y = 'x*Ser(seq(300),'x); 0 == 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x
\\ Gheorghe Coserea, Jul 07 2018

b(n) = sum(k=0, n\2, n!/(k!^2 * (n - 2*k)!)); \\ A002426
a(n) = 2*sum(k=0, n-1, binomial(2*n, k) * 3^k * b(n-k))/(n+1);
vector(18, n, a(n)) \\ Gheorghe Coserea, Dec 26 2018

From Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005: (Start)
G.f.: ((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x) where R=sqrt(1-12*x);
a(n) ~ sqrt(3/2)*12^n/(n^(5/4)*GAMMA(3/4)). (End)
From Gheorghe Coserea, Dec 26 2018: (Start)
a(n) = (2/(n+1)) * Sum_{k=0..n-1} binomial(2*n, k) * 3^k * A002426(n-k).
G.f. y=A(x) satisfies:
0 = 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x.
0 = x*(4*x + 1)*(12*x - 1)^3*y'''' + 4*(132*x^2 + 19*x - 1)*(12*x - 1)^2*y''' + 12*(1476*x^2 + 60*x - 11)*(12*x - 1)*y'' + 72*(2016*x^2 - 117*x - 4)*y' + 648*(16*x - 1)*y.
(End)

Reference gives 20 terms
Description corrected May 15 1997, thanks to Jean-Francois Beraud
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005

A343089 Number of nonseparable rooted toroidal maps with n edges.

Original entry on oeis.org

1, 8, 59, 420, 2940, 20384, 140479, 964184, 6598481, 45059872, 307197620, 2091615760, 14226362200, 96680047568, 656559634503, 4456100344560, 30228597199443, 204971912361512, 1389342336011059, 9414200925647540, 63772600432265968, 431892497914345472

Offset: 2

Andrew Howroyd, Apr 04 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..100

Row sums of A342989.

Cf. A006300.

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A238359 Number of genus-9 rooted maps with n edges.

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A238360 Number of genus-10 rooted maps with n edges.

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A006386 Number of sensed genus 1 maps with n edges.

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A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

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A006387 Number of unsensed genus 1 maps with n edges.

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A007137 Number of rooted maps with n edges on the projective plane.

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A343089 Number of nonseparable rooted toroidal maps with n edges.

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