A238355 -id:A238355 - cp's OEIS frontend

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

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
```

A269925 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 5.

Original entry on oeis.org

59520825, 4304016990, 4304016990, 158959754226, 354949166565, 158959754226, 4034735959800, 14805457339920, 14805457339920, 4034735959800, 79553497760100, 420797306522502, 691650582088536, 420797306522502, 79553497760100, 1302772718028600, 9220982517965400, 21853758736216200, 21853758736216200, 9220982517965400, 1302772718028600

Offset: 10

Gheorghe Coserea, Mar 15 2016

Row n contains n-9 terms.

			Triangle starts:
n\f  [1]             [2]             [3]             [4]
[10] 59520825;
[11] 4304016990,     4304016990;
[12] 15895975422,    354949166565,   158959754226;
[13] 4034735959800,  14805457339920, 14805457339920, 4034735959800;
[14] ...

Gheorghe Coserea, Rows n = 10..210, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Rooted maps of genus 5 with n edges and f faces for 1<=f<=10: A288281 f=1, A288282 f=2, A288283 f=3, A288284 f=4, A288285 f=5, A288286 f=6, A288287 f=7, A288288 f=8, A288289 f=9, A288290 f=10.
Row sums give A238355 (column 5 of A269919).
Cf. A035309, A269921, A269922, A269923, A269924, A270406, A270407, A270408, A270409, A270410, A270411, A270412.

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3)(2n-2)(2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1)(2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
Table[Q[n, f, 5], {n, 10, 15}, {f, 1, n-9}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

N = 15; G = 5; gmax(n) = min(n\2, G);
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

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)

A239918 Number of unrooted maps with n edges of (orientable) genus 5.

Original entry on oeis.org

2976853, 391288854, 28036387466, 1449247494892, 60441165724160, 2158501051914340, 68463726004852884, 1976314846820429680, 52825750657523709792, 1324265997531577820388, 31439565426089264422698, 712298211293218414835136

Offset: 10

Alain Giorgetti, Mar 29 2014

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

Column k=5 of A379438.
Cf. A238355 (rooted), A297881 (unsensed).
Cf. A006386, A104595, A104596, A215019.

A297881 Number of unsensed genus 5 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1491629, 195728778, 14019733828, 724646387874, 30220873171570, 1079253898643492, 34231899372185491, 988157793188200998, 26412878913430197293, 662133032168309300424, 15719783014093104131694

Offset: 0

Evgeniy Krasko, Jan 07 2018

nonn

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.

Column k=5 of A379439.
Cf. A238355 (rooted), A239918 (sensed).
Cf. A006385, A006387, A214814, A214815, A214816, A297880.

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

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A269925 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 5.

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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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A239918 Number of unrooted maps with n edges of (orientable) genus 5.

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A297881 Number of unsensed genus 5 maps with n edges.

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