A321710 -id:A321710 - cp's OEIS frontend

A318104 Number of genus 4 rooted hypermaps with n darts.

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.

A321705 Number of genus 5 rooted hypermaps with n darts.

Original entry on oeis.org

604800, 57170880, 2936606400, 108502598960, 3225186125460, 81861294718764, 1840409325096500, 37558997857897164, 708015469597497732, 12488421105878928700, 208161512148250424484, 3304395638081490531324, 50267199680265668419244, 736516493829967530909204, 10437808798822929984593100

Offset: 11

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 11..111
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 7
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 5 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17));
};
seq(15)

G.f.: y*(y - 1)^11*(13150*y^19 - 315600*y^18 + 6947865*y^17 - 70489470*y^16 + 569637816*y^15 - 3253135788*y^14 + 14658702716*y^13 - 51696766668*y^12 + 146255446788*y^11 - 332779761068*y^10 + 610739916966*y^9 - 900544355928*y^8 + 1057440629016*y^7 - 973453624356*y^6 + 685359139356*y^5 - 355019010868*y^4 + 127180243662*y^3 - 28342783668*y^2 + 3224985513*y - 120590634)/(4*(y - 2)^22*(y + 1)^17), where y=A000108(2*x).

A321706 Number of genus 6 rooted hypermaps with n darts.

Original entry on oeis.org

68428800, 8099018496, 511859777472, 22925949056640, 815521082030784, 24494440792190400, 645212095792089220, 15292175926873102956, 332150183310464271324, 6702637985834037183508, 126995200843857803023176, 2278149500006567629947864, 38954050134978747926573016

Offset: 13

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 13..113
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 8
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 6 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(-y*(y - 1)^13*(1080091*y^24 - 32402730*y^23 + 889296813*y^22 - 11575684382*y^21 + 120636055215*y^20 - 908735922846*y^19 + 5491340556019*y^18 - 26587756725282*y^17 + 105914199493428*y^16 - 349844034215428*y^15 + 966356094916770*y^14 - 2240740995310188*y^13 + 4368032453176430*y^12 - 7149882085566108*y^11 + 9789363335577126*y^10 - 11134972065337540*y^9 + 10413235525450707*y^8 - 7887398782084338*y^7 + 4736927774219617*y^6 - 2188131419800854*y^5 + 743586620967027*y^4 - 173682661266854*y^3 + 24974862235959*y^2 - 1816988020602*y + 43470403150)/(4*(y - 2)^27*(y + 1)^21));
};
seq(13)

G.f.: -y*(y - 1)^13*(1080091*y^24 - 32402730*y^23 + 889296813*y^22 - 11575684382*y^21 + 120636055215*y^20 - 908735922846*y^19 + 5491340556019*y^18 - 26587756725282*y^17 + 105914199493428*y^16 - 349844034215428*y^15 + 966356094916770*y^14 - 2240740995310188*y^13 + 4368032453176430*y^12 - 7149882085566108*y^11 + 9789363335577126*y^10 - 11134972065337540*y^9 + 10413235525450707*y^8 - 7887398782084338*y^7 + 4736927774219617*y^6 - 2188131419800854*y^5 + 743586620967027*y^4 - 173682661266854*y^3 + 24974862235959*y^2 - 1816988020602*y + 43470403150)/(4*(y - 2)^27*(y + 1)^21), where y=A000108(2*x).

A321707 Number of genus 7 rooted hypermaps with n darts.

Original entry on oeis.org

10897286400, 1560315052800, 117805728533760, 6234567636407040, 259518044572234560, 9042873557130178560, 274213957041780607040, 7429773263737371426000, 183321599847270732775284, 4178263675886440605897708, 88944569813776527012125700, 1783998282211666387087167804

Offset: 15

Gheorghe Coserea, Nov 17 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 15..115
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 7 of A321710.

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec(y*(y - 1)^15*(65934428*y^29 - 2373639408*y^28 + 78065287272*y^27 - 1239982167455*y^26 + 15824435903631*y^25 - 147973440711114*y^24 + 1121119206347992*y^23 - 6897821674771866*y^22 + 35411504747483046*y^21 - 153195355715747532*y^20 + 564430051160248776*y^19 - 1782487694948370405*y^18 + 4847762532445875077*y^17 - 11385796789588847190*y^16 + 23121306258262005552*y^15 - 40583304876425900476*y^14 + 61456131914028384816*y^13 - 80013191284851054552*y^12 + 89093563965211655640*y^11 - 84217909084706439705*y^10 + 66908561791095714729*y^9 - 44081770356535124534*y^8 + 23656775380308201720*y^7 - 10093706157332984394*y^6 + 3311499493215102574*y^5 - 796565568171137388*y^4 + 130914104820311544*y^3 - 13151257876934851*y^2 + 664822027105443*y - 11057829184170)/(2*(y - 2)^32*(y + 1)^25));
};
seq(12)

G.f.: y*(y - 1)^15*(65934428*y^29 - 2373639408*y^28 + 78065287272*y^27 - 1239982167455*y^26 + 15824435903631*y^25 - 147973440711114*y^24 + 1121119206347992*y^23 - 6897821674771866*y^22 + 35411504747483046*y^21 - 153195355715747532*y^20 + 564430051160248776*y^19 - 1782487694948370405*y^18 + 4847762532445875077*y^17 - 11385796789588847190*y^16 + 23121306258262005552*y^15 - 40583304876425900476*y^14 + 61456131914028384816*y^13 - 80013191284851054552*y^12 + 89093563965211655640*y^11 - 84217909084706439705*y^10 + 66908561791095714729*y^9 - 44081770356535124534*y^8 + 23656775380308201720*y^7 - 10093706157332984394*y^6 + 3311499493215102574*y^5 - 796565568171137388*y^4 + 130914104820311544*y^3 - 13151257876934851*y^2 + 664822027105443*y - 11057829184170)/(2*(y - 2)^32*(y + 1)^25), where y=A000108(2*x).

A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.

Original entry on oeis.org

1, 3, 6, 1, 20, 6, 60, 33, 4, 291, 285, 48, 1310, 2115, 708, 30, 6975, 16533, 9807, 1155, 37746, 126501, 119436, 29910, 900, 215602, 972441, 1355400, 601364, 58032, 1262874, 7451679, 14561360, 10260804, 2112300, 54990, 7611156, 57167260, 150429819, 156469887, 57017238, 4764654

Offset: 1

Paawan Jethva, Jun 22 2025

Note that Sum_{g>=0} D(n,g) gives A057005 which is the number of dessins d'enfants with n edges (as one would hope).
We get a new genus every two edges.
n=7 is the first time we have more dessins of genus 1 than genus 0.

			Triangle D(n,g) begins:
   n\g    0      1      2      3      4      ...
   1      1
   2      3
   3      6      1
   4      20     6
   5      60     33     4
   6      291    285    48
   7      1310   2115   708    30
   8      6975   16533  9807   1155
   9      37746  126501 119436 29910  900
   ...

Paawan Jethva, Exploring the Euler Characteristics of Dessins d’Enfants, 2023, page 15.
Ján Karabáš, Enumeration of actions of cyclic groups on orientable surfaces.
A. Mednykh and R. Nedela, 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), tables 2-8 and theorems 8-12. Theorem 12 has typos; the corrected formula can be inferred from Karabáš's tables.

Cf. A057005.

Columns: A090371, A118094, A214819, A214820, A356694. A321710 is the rooted version.

Rows 10-11 from Andrei Zabolotskii, Jun 28 2025

A321708 Number of genus 8 rooted hypermaps with n darts.

Original entry on oeis.org

2324754432000, 392578737868800, 34628437757170944, 2123123613378727680, 101617019078765734656, 4043718573909964925184, 139173710438785042157056, 4255877128208539192614720, 117904584208478764267020480, 3002991800410924564910152160, 71123951019277304732006170020

Offset: 17

Gheorghe Coserea, Nov 17 2018

nonn

Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 8 of A321710.

A321709 Number of genus 9 rooted hypermaps with n darts.

Original entry on oeis.org

640237370572800, 125042114810880000, 12663309118470912000, 885531242655070387200, 48050907406540671926016, 2155959161260047868965120, 83244478130310272277600000, 2842577637658581528366266112, 87562415111185051655245105920, 2469899876154506406497571909952

Offset: 19

Gheorghe Coserea, Nov 17 2018

nonn

Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Column 9 of A321710.

cp's OEIS Frontend

A318104 Number of genus 4 rooted hypermaps with n darts.

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A321705 Number of genus 5 rooted hypermaps with n darts.

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A321706 Number of genus 6 rooted hypermaps with n darts.

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A321707 Number of genus 7 rooted hypermaps with n darts.

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A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.

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A321708 Number of genus 8 rooted hypermaps with n darts.

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A321709 Number of genus 9 rooted hypermaps with n darts.

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