A214817 -id:A214817 - cp's OEIS frontend

A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496

Offset: 1

Gheorghe Coserea, Nov 17 2018

Row n contains floor((n+1)/2) = A008619(n-1) terms.

			Triangle starts:
n\k  [0]       [1]        [2]         [3]         [4]        [5]
[1]  1;
[2]  3;
[3]  12,       1;
[4]  56,       15;
[5]  288,      165,       8;
[6]  1584,     1611,      252;
[7]  9152,     14805,     4956,       180;
[8]  54912,    131307,    77992,      9132;
[9]  339456,   1138261,   1074564,    268980,     8064;
[10] 2149888,  9713835,   13545216,   6010220,    579744;
[11] 13891584, 81968469,  160174960,  112868844,  23235300,  604800;
[12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
[13] ...

Gheorghe Coserea, Rows n = 1..42, flattened
Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Columns k=0..9 give: A000257 (k=0), A118093 (k=1), A214817 (k=2), A214818 (k=3), A318104 (k=4), A321705 (k=5), A321706 (k=6), A321707 (k=7), A321708 (k=8), A321709 (k=9).
Row sums give A003319(n+1).
Cf. A008619, A060593.

l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}];
m1[f_,n_] := Sum[(i-1)t[j]t[i-j]D[f,t[i-1]] + j(i-j)t[i+1]D[f,t[j],t[i-j]], {i,2,n},{j,i-1}];
ff[1] = x^2 t[1];
ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1],n] + m1[ff[n-1],n] + Sum[t[i+1]j(i-j)D[ff[k],t[j]]D[ff[n-1-k],t[i-j]], {i,2,n-1},{j,i-1},{k,n-2}]) / n;
row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]];
Table[row[n], {n,14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *)

L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
M1(f, N) = {
  sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
      j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
};
F(N) = {
  my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
  f[1] = u*v*t[1];
  for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
    sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
    j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
    f[n] /= n);
  f;
};
seq(N) = {
  my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
     g=vector(#v, n, Polrev(Vec(n * v[n]))));
  apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
};
concat(seq(14))

A000257(n)=T(n,0), A118093(n)=T(n,1), A214817(n)=T(n,2), A214818(n)=T(n,3), A060593(n)=T(2*n+1,n)=(2*n)!/(n+1), A003319(n+1)=Sum_{k=0..floor((n-1)/2)} T(n,k).

A118093 Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).

Original entry on oeis.org

1, 15, 165, 1611, 14805, 131307, 1138261, 9713835, 81968469, 685888171, 5702382933, 47168678571, 388580070741, 3190523226795, 26124382262613, 213415462218411, 1740019150443861, 14162920013474475, 115112250539595093, 934419385591442091, 7576722323539318101

Offset: 3

Valery A. Liskovets, Apr 13 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..500
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 3
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.

Cf. A000108, A000257, A118094, A214817, A214818.

[&+[(2^k*(4^(n-2-k)-1)*Binomial(n+k, k))/3 : k in [0..n-3]]: n in [3..25]]; // Vincenzo Librandi, Sep 16 2018

Table[Sum[2^k (4^(n - 2 - k) - 1) Binomial[n+k, k] / 3, {k, 0, n-3}], {n, 3, 25}] (* Vincenzo Librandi, Sep 16 2018 *)

a(n) = sum(k=0, n-3, 2^k*(4^(n-2-k)-1)*binomial(n+k, k))/3; \\ Michel Marcus, Dec 11 2014

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec((y - 1)^3/(4*(y - 2)^2*(y + 1)));
};
seq(21) \\ Gheorghe Coserea, Nov 06 2018

Conjecture: +n*(5*n-17)*a(n) -15*(n-1)*(5*n-16)*a(n-1) +12*(20*n^2-103*n+140)*a(n-2) +32*(5*n-12)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 05 2018
G.f.: (1 - 7*x + 4*x^2 - (1 - 3*x)*sqrt(1 - 8*x))/(8*(1 + x)*(1 - 8*x)); equivalently, the g.f. can be rewritten as (y - 1)^3/(4*(y - 2)^2*(y + 1)), where y=G(2*x) with G the g.f. of A000108. - Gheorghe Coserea, Nov 06 2018
a(n) ~ 2^(3*n - 4) / 3 * (1 - 10/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Nov 06 2018

More terms from Michel Marcus, Dec 11 2014

A214818 Number of genus 3 rooted hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 180, 9132, 268980, 6010220, 112868844, 1877530740, 28540603884, 404562365316, 5422718644920, 69428442576136, 855504181649448, 10204459810035768, 118364711625485256, 1340006035830921720, 14850353930248138104, 161502853638370415864, 1727146533728893094604

Offset: 1

N. J. A. Sloane, Aug 01 2012

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..107 (corrected by _Georg Fischer_, Jan 20 2019)
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 5
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
Peter G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Cf. A000108, A000257, A118093, A214817, A003319.

DeleteCases[CoefficientList[Series[# (# - 1)^7*(5 #^9 - 60 #^8 + 675 #^7 - 2947 #^6 + 10005 #^5 - 20235 #^4 + 28297 #^3 - 23937 #^2 + 11418 # - 1781)/(2 (# - 2)^12*(# + 1)^9) &[(1 - Sqrt[1 - 8 x])/(4 x)], {x, 0, 23}], x], 0] (* Michael De Vlieger, Nov 26 2018 *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(y*(y - 1)^7*(5*y^9 - 60*y^8 + 675*y^7 - 2947*y^6 + 10005*y^5 - 20235*y^4 + 28297*y^3 - 23937*y^2 + 11418*y - 1781)/(2*(y - 2)^12*(y + 1)^9));
};
seq(18) \\ Gheorghe Coserea, Nov 12 2018

G.f.: y*(y - 1)^7*(5*y^9 - 60*y^8 + 675*y^7 - 2947*y^6 + 10005*y^5 - 20235*y^4 + 28297*y^3 - 23937*y^2 + 11418*y - 1781)/(2*(y - 2)^12*(y + 1)^9), where y = C(2*x), C being the g.f. for A000108. - Gheorghe Coserea, Nov 12 2018

a(13)-a(14) by Noam Zeilberger, Sep 16 2018

More terms from Gheorghe Coserea, Nov 11 2018

A318104 Number of genus 4 rooted hypermaps with n darts.

Original entry on oeis.org

8064, 579744, 23235300, 684173164, 16497874380, 344901105444, 6471056247920, 111480953909328, 1792031518697232, 27197316623478960, 393207192141924744, 5453210050430783640, 72949244341257096792, 945523594111460363208, 11918067649004916470640, 146538779626167833263888, 1762112462707129510538640

Offset: 9

Gheorghe Coserea, Nov 12 2018

nonn

Column k = 4 of A321710.

a(n) = 0 for n < 9. - N. J. A. Sloane, Dec 24 2018

			A(x) = 8064*x^9 + 579744*x^10 + 23235300*x^11 + 684173164*x^12 + ...

Gheorghe Coserea, Table of n, a(n) for n = 9..109
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 6
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Cf. A000257, A118093, A214817, A214818, A321710.

y = (1 - Sqrt[1 - 8 x])/(4 x);
gf = -y (y-1)^9 (262 y^14 - 4716 y^13 + 78327 y^12 - 569134 y^11 + 3266910 y^10 - 12675726 y^9 + 37548087 y^8 - 82680972 y^7 + 137674842 y^6 - 170295272 y^5 + 152918277 y^4 - 94811622 y^3 + 37127810 y^2 - 7566846 y + 505869)/(4 (y-2)^17 (y+1)^13);
Drop[CoefficientList[gf + O[x]^26, x], 9] (* Jean-François Alcover, Feb 07 2019, from PARI *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(-y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13));
};
seq(17)

G.f.: -y*(y - 1)^9*(262*y^14 - 4716*y^13 + 78327*y^12 - 569134*y^11 + 3266910*y^10 - 12675726*y^9 + 37548087*y^8 - 82680972*y^7 + 137674842*y^6 - 170295272*y^5 + 152918277*y^4 - 94811622*y^3 + 37127810*y^2 - 7566846*y + 505869)/(4*(y - 2)^17*(y + 1)^13), where y = C(2*x), C being the g.f. for A000108.

cp's OEIS Frontend

A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

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A118093 Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).

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A214818 Number of genus 3 rooted hypermaps with n darts.

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A318104 Number of genus 4 rooted hypermaps with n darts.

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