A057005 -id:A057005 - cp's OEIS frontend

A110143 Row sums of triangle A110141.

1, 1, 4, 11, 43, 161, 901, 5579, 43206, 378360, 3742738, 40853520, 488029621, 6323154547, 88308425755, 1322120265238, 21122364398761, 358647945023885, 6449299885654827, 122436442904193940, 2447046870232798369, 51358050784584629338, 1129314001779283063606

Offset: 0

Paul D. Hanna, Jul 14 2005

nonn

Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). There are A000041(n) terms in row n of triangle A110141.
Also, number of orbits of Sym(n)^2 where Sym_n acts by conjugation.  Compare the MathOverflow discussion, also Bogaerts-Dukes 2014, and A241584, A241585. - Peter J. Dukes, May 12 2014
Number of isomorphism classes of n-fold coverings of a connected graph with circuit rank 2 [Kwak and Lee]. - Álvar Ibeas, Mar 25 2015

P. A. MacMahon, The expansion of determinants and permanents in terms of symmetric functions, in Proc. ICM Toronto (ed. J. C. Fields), Toronto University Press, 1924, vol 1, 319-330.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. Appears to contain this sequence in Table 2. [Added by N. J. A. Sloane, Nov 12 2009]

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mathieu Bogaerts and Peter Dukes, Semidefinite programming for permutation codes, Discrete Math. 326 (2014), 34--43. MR3188985.
Nicholas Dub, Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
J. B. Geloun, S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv:1307.6490 [hep-th], 2013.
Joseph Ben Geloun, Sanjaye Ramgoolam, All-orders asymptotics of tensor model observables from symmetries of restricted partitions, arXiv:2106.01470 [hep-th], Jun 02 2021.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n [From Peter Dukes, May 12 2014]
Igor Pak, Greta Panova, Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.

Cf. A110141, A000041, A057005, A096161, A241584, A241585, A279819.

Third column of A160449.

# Using a function from Alois P. Heinz in A279038:
b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(i<1, [], [seq(map(x-> x*i^j*j!,
      b(n-i*j, i-1))[], j=0..n/i)]))
    end:
seq(add(i, i=b(n$2)), n=0..22); # Peter Luschny, Dec 19 2016

Table[Total[Apply[Times, Tally[#]/.{a_Integer,b_}->a^b b!]& /@ IntegerPartitions[n]],{n,0,21}] (* Wouter Meeussen, Oct 17 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Flatten[ Table[ Map[ #*i^j*j!&, b[n-i*j, i-1]], {j, 0, n/i}]]]]; Table[Sum[i, {i, b[n, n]}], {n, 0, 22}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
nmax = 25; CoefficientList[Series[Product[Sum[k!*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2019 *)
m = 30; CoefficientList[Series[Product[-Gamma[0, -1/(x^j*j)] * Exp[-1/(x^j*j)], {j, 1, m}] / (x^(m*(m + 1)/2)*m!), {x, 0, m}], x] (* Vaclav Kotesovec, Dec 07 2020 *)

def A110143(n):
    return sum(p.aut() for p in Partitions(n))
[A110143(n) for n in range(9)]
# Álvar Ibeas, Mar 26 2015

G.f.: B(x)*B(2*x^2)*B(3*x^3)*..., where B(x) is g.f. of A000142. - Vladeta Jovovic, Feb 18 2007

a(n) ~ n! * (1 + 2/n^2 + 5/n^3 + 23/n^4 + 106/n^5 + 537/n^6 + 3143/n^7 + 20485/n^8 + 143747/n^9 + 1078660/n^10), for coefficients see A279819. - Vaclav Kotesovec, Mar 16 2015

A057004 Array T(n,k) = number of conjugacy classes of subgroups of index k in free group of rank n, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 41, 26, 1, 1, 31, 235, 604, 97, 1, 1, 63, 1361, 14120, 13753, 624, 1, 1, 127, 7987, 334576, 1712845, 504243, 4163, 1, 1, 255, 47321, 7987616, 207009649, 371515454, 24824785, 34470, 1, 1, 511, 281995, 191318464

Offset: 1

N. J. A. Sloane, Sep 09 2000

			Array T(n,k) begins:
1 1 1 1 1 1 1 ...
1 3 7 26 97 624 4163 ...
1 7 41 604 13753 504243 ...
1 15 235 14120 1712845 ...

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

M. Hofmeister, A Note on Counting Connected Graph Covering Projections, SIAM J. Discrete Math., 11 (1998), 286-292. See page 291 Table 4.3.
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Rows, columns, main diagonal give A057005-A057013, A160871.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A214819 Number of genus 2 sensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 4, 48, 708, 9807, 119436, 1355400, 14561360, 150429819, 1506841872, 14732613116, 141226638540, 1331912032173, 12390368538412, 113927616087252, 1037080582036632, 9358430685657218, 83804192879934456, 745394788170961932, 6590038606472968276

Offset: 1

N. J. A. Sloane, Aug 01 2012

nonn

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), table 4.
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.

Cf. A090371, A118094, A214820, A057005, A380453.

hO[d_, , ] := 0 /; !IntegerQ@d;
hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
h[0][m_] := 3 2^(m-1) Binomial[2m,m] / ((m+1)(m+2));
h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k,k], {k,0,d-3}] / 3;
h[2][d_] := Coefficient[-# (# - 1)^5 (#^4 - 6 #^3 + 36 #^2 - 50 # + 51) / (4 (# - 2)^7 (# + 1)^5) &[(1-Sqrt[1-8x])/(4x)+O[x]^(d+1)], x, d];
a2[d_] := (h[2][d] + 4hO[d/2,1,{2}] + hO[d/2,0,{6}] + 6hO[d/3,0,{0,4}] + 2hO[d/4,0,{2,0,2}] + 12hO[d/5,0,{0,0,0,3}] + 2hO[d/6,0,{2,2}] + 2hO[d/6,0,{0,1,0,0,2}] + 4hO[d/8,0,{1,0,0,0,0,0,2}] + 4hO[d/10,0,{1,0,0,1,0,0,0,0,1}]) / d;
Table[a2[n], {n, 23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 8 *)

Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025

A214820 Number of genus 3 sensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 30, 1155, 29910, 601364, 10260804, 156469887, 2195431068, 28897471080, 361514582340, 4339280187364, 50323775391144, 566914469842923, 6229721664499224, 67000302262906866, 707159710965012834, 7341038807584085816, 75093327553430134548

Offset: 1

N. J. A. Sloane, Aug 01 2012

nonn

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), 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.

Cf. A090371, A118094, A214819, A057005, A380453.

hO[d_, , ] := 0 /; !IntegerQ@d;
hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
h[0][m_] := 3 2^(m-1) Binomial[2m, m] / ((m+1)(m+2));
h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k, k], {k, 0, d-3}] / 3;
h[2][d_] := Coefficient[-# (# - 1)^5 (#^4 - 6 #^3 + 36 #^2 - 50 # + 51) / (4 (# - 2)^7 (# + 1)^5) &[(1-Sqrt[1-8x])/(4x) + O[x]^(d+1)], x, d];
h[3][d_] := Coefficient[# (# - 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[1-8x])/(4x) + O[x]^(d+1)],x,d];
a3[d_] := (h[3][d] + 15h[2][d/2] + 4hO[d/2,1,{4}] + hO[d/2,0,{8}] + 18hO[d/3,1,{2}] + 10hO[d/3,0,{5}] + 12hO[d/4,1,{2}] + 2hO[d/4,0,{3,2}] + 8hO[d/4,0,{4}] + 2hO[d/6,0,{1,2,1}] + 2hO[d/6,0,{2,2}] + 30hO[d/7,0,{3}] + 8hO[d/8,0,{1,2}] + 12hO[d/9,0,{1,2}] + 4hO[d/12,0,{1,2}] + 4hO[d/12,0,{1,1,1}] + 6hO[d/14,0,{1,1,1}]) / d;
Table[a3[n],{n,23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 9 *)

Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025

A214821 Number of genus 0 unsensed hypermaps with n darts.

Original entry on oeis.org

1, 3, 6, 20, 57, 240, 954, 4566, 22641, 121823, 683307, 4004055

Offset: 1

N. J. A. Sloane, Aug 01 2012

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

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A214823 Number of genus 2 unsensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 4, 39, 456, 5554, 63378, 698568, 7391499, 75807708

Offset: 1

N. J. A. Sloane, Aug 01 2012

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

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A215017 Number of genus 3 unsensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 25, 678, 15867, 307880, 5180472, 78573507

Offset: 1

N. J. A. Sloane, Aug 01 2012

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

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A215018 Number of unsensed hypermaps with n darts and any genus.

Original entry on oeis.org

1, 3, 7, 26, 91, 490, 2785, 20434, 171579, 1671193, 18192737, 218487504

Offset: 1

N. J. A. Sloane, Aug 01 2012

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

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

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

A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

Original entry on oeis.org

1, 7, 41, 235, 1361, 7987, 47321, 281995, 1685921, 10096867, 60524201, 362972155, 2177309681, 13062280147, 78368930681, 470199300715, 2821152888641, 16926788453827, 101560343826761, 609360901747675, 3656161925798801, 21936961098633907, 131621735219132441

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

Starting at a(2), consider that 2/3 - 1/2 = 1/6 with 1+6=7=a(2); 8/9 - 3/4 = 5/36 with 5+36=41=a(3); 26/27 - 7/8=19/216 with 19+216=235=a(4); 80/81 - 15/16=65/1296 with 65+1296=1361=a(5) and so forth. The numerators starting at a(3) are 5,19,65,211,665,2059,6305,... (see A001047) with 19 mod 5=4, 65 mod 19=8, 211 mod 65=16, 665 mod 211=32, 2059 mod 665=64, 6305 mod 2059=128, and so forth for higher powers of 2. - J. M. Bergot, May 09 2015

In other words, let f(n) = (3^(n-1)-1)/3^(n-1) - (2^(n-1)-1)/2^(n-1), then for n>=1 a(n) = numerator(f(n)) + denominator(f(n)). - Michel Marcus, May 29 2015

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A057004, A057005, A057006, A057007, A057008, A057010, A057011, A057012, A057013.

[6^(n-1)+3^(n-1)-2^(n-1): n in [1..30]] /* or */ I:=[1,7,41]; [n le 3 select I[n] else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 12 2015

Table[6^(n-1)+3^(n-1)-2^(n-1),{n,25}] (* or *) LinearRecurrence[ {11,-36,36},{1,7,41},25] (* Harvey P. Dale, Nov 24 2011 *)
CoefficientList[Series[(1 - 4 x)/((1 - 2 x) (1 - 3 x) (1 - 6 x)), {x, 0, 33}], x] (* Vincenzo Librandi, May 12 2015 *)

PARI
```
a(n)=if(n<0,0,6^(n-1)+3^(n-1)-2^(n-1))
```

G.f.: x(1-4x)/((1-2x)(1-3x)(1-6x)). a(n) = 6^(n-1)+3^(n-1)-2^(n-1).
E.g.f.: e^(6*x)+e^(3*x)-e^(2*x). [Mohammad K. Azarian, Jan 16 2009]
a(1)=1, a(2)=7, a(3)=41, a(n) = 11*a(n-1)-36*a(n-2)+36*a(n-3). [Harvey P. Dale, Nov 24 2011]

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

cp's OEIS Frontend

A110143 Row sums of triangle A110141.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A057004 Array T(n,k) = number of conjugacy classes of subgroups of index k in free group of rank n, read by antidiagonals.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A214819 Number of genus 2 sensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A214820 Number of genus 3 sensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A214821 Number of genus 0 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

A214823 Number of genus 2 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

A215017 Number of genus 3 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

A215018 Number of unsensed hypermaps with n darts and any genus.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

Views

Author

Keywords

Comments

References