A215402 -id:A215402 - 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
```

A269924 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 4.

Original entry on oeis.org

225225, 12317877, 12317877, 351683046, 792534015, 351683046, 7034538511, 26225260226, 26225260226, 7034538511, 111159740692, 600398249550, 993494827480, 600398249550, 111159740692, 1480593013900, 10743797911132, 25766235457300, 25766235457300, 10743797911132, 1480593013900, 17302190625720, 160576594766588, 517592962672296, 750260619502310, 517592962672296, 160576594766588, 17302190625720

Offset: 8

Gheorghe Coserea, Mar 15 2016

Row n contains n-7 terms.

			Triangle starts:
n\f  [1]           [2]           [3]           [4]
[8]  225225;
[9]  12317877,     12317877;
[10] 351683046,    792534015,    351683046;
[11] 7034538511,   26225260226,  26225260226,  7034538511;
[12] ...

Gheorghe Coserea, Rows n = 8..208, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Columns f=1-10 give: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Cf. A035309, A269921, A269922, A269923, A269925, A270406, A270407, A270408, A270409, A270410, A270412.
Row sums give A215402 (column 4 of A269919).

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, 4], {n, 8, 14}, {f, 1, n-7}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

N = 14; G = 4; 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))))

A215019 Number of unrooted maps with n edges of (orientable) genus 4.

Original entry on oeis.org

14118, 1369446, 74803564, 3023693380, 100692692173, 2922359760376, 76471600288836, 1845089145736960, 41694584320696782, 892580319444417876, 18258463136626650660, 359279139700128276168, 6836732826365623258492, 126347598971804884131800, 2275643837092089686415858

Offset: 8

N. J. A. Sloane, Aug 01 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 8..45
Alexander Mednykh, Alain Giorgetti, Enumeration of genus four maps by number of edges. Ars Mathematica Contemporanea 4 (2011), 351-361. MR2842107
A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, preprint (submitted to J. Combin. Th. B).
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=4 of A379438.
Cf. A215402 (rooted), A297880 (unsensed).
Cf. A006386, A104595, A104596 (genus 3), A239918 (genus 5), A239919 (genus 6).

a(12) onwards added by Andrew Howroyd, Jan 18 2025

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)

A297880 Number of unsensed genus 4 maps with n edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 7258, 688976, 37466297, 1512650776, 50355225387, 1461269893538, 38236656513725, 922552326544030, 20847359639841664, 446290728182323620, 9129236228868478458, 179639607187998993180, 3418366706444416598777

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=4 of A379439.
Cf. A215402 (rooted), A215019 (sensed).
Cf. A006385, A006387, A214814, A214815, A214816.

a(20) added by Andrew Howroyd, Jan 20 2025

cp's OEIS Frontend

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

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A269924 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 4.

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A215019 Number of unrooted maps with n edges of (orientable) genus 4.

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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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A297880 Number of unsensed genus 4 maps with n edges.

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