A160449 -id:A160449 - 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

A074528 a(n) = 2^n + 3^n + 6^n.

Original entry on oeis.org

3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251

Offset: 0

Robert G. Wilson v, Aug 23 2002

From Álvar Ibeas, Mar 24 2015: (Start)
Number of isomorphism classes of 3-fold coverings of a connected graph with circuit rank n+1 [Kwak and Lee].
Number of orbits of the conjugacy action of Sym(3) on Sym(3)^(n+1) [Kwak and Lee, 2001].
(End)

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. [Added by N. J. A. Sloane, Nov 12 2009]

Hakan Icoz, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.
A246985 is essentially identical.
Third row of A160449, shifted.

[2^n + 3^n + 6^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 6^n, {n, 0, 20}]
LinearRecurrence[{11,-36,36},{3,11,49},30] (* Harvey P. Dale, May 02 2016 *)

a(n)=2^n+3^n+6^n \\ Charles R Greathouse IV, Oct 07 2015

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x)+1/(1-3*x)+1/(1-6*x).
E.g.f.: exp(2*x) + exp(3*x) + exp(6*x). (End)
a(n) = 11*a(n-1) - 36*a(n-2) + 36*a(n-3). - Wesley Ivan Hurt, Aug 21 2020

A160450 Expansion of (1-34x+276x^2-584x^3)/((1-3x)(1-4x)(1-8x)(1-24x)).

Original entry on oeis.org

1, 5, 43, 681, 14491, 336465, 7997683, 191374041, 4588603531, 110092229025, 2641942301923, 63404456863401, 1521689741669371, 36520416189619185, 876488888356983763, 21035724521756752761, 504857318142580028011, 12116575072428716250945, 290797797234516859979203, 6979147097598917713826121

Offset: 0

N. J. A. Sloane, Nov 15 2009

Number of isomorphism classes of 4-fold coverings of a connected graph with Betti number n. [Kwak and Lee]

Number of orbits of the conjugacy action of Sym(4) on Sym(4)^n [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 0..500
M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv preprint arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (39,-428,1728,-2304).

Fourth row of A160449.

Table[3^(n - 1) + 2*4^(n - 1) + 8^(n - 1) + 24^(n - 1), {n, 0, 19}] (* Michael De Vlieger, Mar 24 2015 *)
LinearRecurrence[{39,-428,1728,-2304},{1,5,43,681},20] (* Harvey P. Dale, Feb 06 2017 *)

Vec((1-34*x+276*x^2-584*x^3)/((1-3*x)*(1-4*x)*(1-8*x)*(1-24*x)) + O(x^30)) \\ Michel Marcus, Jan 14 2016

G.f.: (1-34*x+276*x^2-584*x^3)/((1-3*x)*(1-4*x)*(1-8*x)*(1-24*x)).

a(n) = 3^(n-1) + 2*4^(n-1) + 8^(n-1) + 24^(n-1). - Álvar Ibeas, Mar 24 2015

Entry revised by N. J. A. Sloane, Sep 15 2014

A152612 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 3.

Original entry on oeis.org

1, 8, 49, 681, 14721, 524137, 25471105, 1628116890, 131789656610, 13174980291658, 1593894406662866, 229496526010111571, 38782290669508033003, 7600987633299112125995, 1710169549495739472301076, 437793904386312274903991653, 126520458075485848061740557461

Offset: 1

N. J. A. Sloane, Nov 12 2009

nonn

Number of orbits of the conjugacy action of Sym(n) on Sym(n)^3 [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 1..60
Joseph Ben Geloun and Sanjaye Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, Annales de l'Institut Henri Poincaré D, Vol. 1, No. 1 (2014), pp. 77-138; arXiv preprint, arXiv:1307.6490 [hep-th], 2013.
Jin Ho Kwak and Jaeun Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.

Euler transform of A057006.

Fourth column of A160449.

A057006 = Import["https://oeis.org/A057006/b057006.txt", "Table"][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[DivisorSum[j, # p[#]&] b[n - j], {j, 1, n}]/n]; b];
a = etr[A057006[[#]]&];
Array[a, 15] (* Jean-François Alcover, Aug 29 2019 *)

[sum(p.aut()**2 for p in Partitions(n)) for n in range(1,8)] # Álvar Ibeas, Mar 24 2015

a(6) and a(7) from Geloun and Ramgoolan (2013) added by N. J. A. Sloane, Nov 21 2013

Name clarified and more terms added by Álvar Ibeas, Mar 24 2015

A160446 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 4.

Original entry on oeis.org

1, 16, 251, 14491, 1730861, 373486525, 128038522439, 65551419139302, 47785761199635528, 47785253957386480534, 63601854214623350663136, 109903723926415382728069729, 241458148813601665905147070195

Offset: 1

N. J. A. Sloane, Nov 12 2009

nonn

Number of orbits of the conjugacy action of Sym(n) on Sym(n)^4 [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 1..60
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.

Fifth column of A160449.

[sum(p.aut()**3 for p in Partitions(n)) for n in range(1,9)] # Álvar Ibeas, Mar 24 2015

Name clarified and more terms added by Álvar Ibeas, Mar 24 2015

A160447 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 5.

Original entry on oeis.org

1, 1, 32, 1393, 336465, 207388305, 268749463729, 645244638648481, 2642912633259448386, 17340131659334061379490, 173401255467914281827442642, 2538767439061885080225425717858, 52643878634689290630033137748571475

Offset: 0

N. J. A. Sloane, Nov 12 2009

nonn

Number of orbits of the conjugacy action of Sym(n) on Sym(n)^5 [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 0..60 [a(0)=1 prepended by _Georg Fischer_, Apr 03 2020]
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.

Sixth column of A160449.

[sum(p.aut()**4 for p in Partitions(n)) for n in range(6)] # Álvar Ibeas, Mar 24 2015

Name clarified and more terms added by Álvar Ibeas, Mar 24 2015

a(0)=1 prepended by F. Chapoton, Mar 15 2020

A160448 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 6.

Original entry on oeis.org

1, 1, 64, 8051, 7997683, 24883501301, 193492277719861, 3252016862827895399, 106562068594917409814646, 6292383326091360022932428280, 629238325608681213686078435061358, 101339461229675874181303485938915652000

Offset: 0

N. J. A. Sloane, Nov 12 2009

nonn

Number of orbits of the conjugacy action of Sym(n) on Sym(n)^6 [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 0..60 [a(0)=1 prepended _Georg Fischer_, Apr 03 2020]
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.

Seventh column of A160449.

[sum(p.aut()**5 for p in Partitions(n)) for n in range(6)] # Álvar Ibeas, Mar 24 2015

Name clarified and more terms added by Álvar Ibeas, Mar 24 2015

a(0)=1 prepended by F. Chapoton, Mar 15 2020

A160454 Number of isomorphism classes of 5-fold coverings of a connected graph with Betti number n.

Original entry on oeis.org

1, 7, 161, 14721, 1730861, 207388305, 24883501301, 2985987361161, 358318118583341, 42998170050574305, 5159780357316368741, 619173642303122852601, 74300837069552376921821, 8916100448264989434407505, 1069932053790827570370392981

Offset: 0

N. J. A. Sloane, Nov 15 2009

Number of orbits of the conjugacy action of Sym(5) on Sym(5)^n [Kwak and Lee, 2001]. - Álvar Ibeas, Mar 24 2015

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.

Álvar Ibeas, Table of n, a(n) for n = 0..450
M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv preprint arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (155,-4670,59440,-374304,1152000,-1382400).

Fifth row of A160449.

LinearRecurrence[{155,-4670,59440,-374304,1152000,-1382400},{1,7,161,14721,1730861,207388305},20] (* Harvey P. Dale, Apr 08 2016 *)

Vec(-(249792*x^5-159200*x^4+36984*x^3-3746*x^2+148*x-1) / ((4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)*(12*x-1)*(120*x-1)) + O(x^100)) \\ Colin Barker, Mar 24 2015

a(n+1) = 4^n + 5^n + 2 * 6^n + 8^n + 12^n + 120^n. - Álvar Ibeas, Mar 24 2015

G.f.: -(249792*x^5-159200*x^4+36984*x^3-3746*x^2+148*x-1) / ((4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)*(12*x-1)*(120*x-1)). - Colin Barker, Mar 24 2015

Name clarified and more terms added by Álvar Ibeas, Mar 24 2015

A176709 Number of isomorphism classes of 6-fold coverings of a connected graph with Betti number n.

Original entry on oeis.org

1, 11, 901, 524137, 373486525, 268749463729, 193492277719861, 139314094050615817, 100306131218514392365, 72220413687258533050849, 51998697816934668649790821, 37439062426374732606941352697, 26956124946902542404909380816605, 19408409961765641983297475951692369

Offset: 0

N. J. A. Sloane, Dec 07 2010

nonn

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.

M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.

Row 6 of A160449.

Name edited, terms a(8) and beyond added by Andrey Zabolotskiy, Sep 02 2022

A246985 Expansion of (1-8x+14x^2)/((1-2x)(1-3x)(1-6*x)).

Original entry on oeis.org

1, 3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251, 3656161927895953, 21936961102828211, 131621735227521049, 789730317205170251, 4738381620767930593

Offset: 0

N. J. A. Sloane, Sep 15 2014

From Álvar Ibeas, Mar 25 2015: (Start)
Number of isomorphism classes of 3-fold coverings of a connected graph with circuit rank n [Kwak and Lee].
Number of orbits of the conjugacy action of Sym(3) on Sym(3)^n [Kwak and Lee].
(End)

J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
A. Prasad, Equivalence classes of nodes in trees and rational generating functions, arXiv:1407.5284 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Apart from first term, same as A074528. Third row of A160449.

[n le 3 select 2*Factorial(n)-1 else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]];

Table[2^(n - 1) + 3^(n - 1) + 6^(n - 1), {n, 0, 30}] (* Bruno Berselli, Mar 25 2015 *)
LinearRecurrence[{11,-36,36},{1,3,11},30] (* Harvey P. Dale, Jan 17 2019 *)

Vec((1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-6*x)) + O(x^30)) \\ Michel Marcus, Jan 14 2016

G.f.: (1-8*x+14*x^2)/((1-2*x)*(1-3*x)*(1-6*x)).
a(n) = 11*a(n-1) - 36*a(n-2) + 36*a(n-3) for n>2. [Bruno Berselli, Mar 25 2015]
a(n) = 2^(n-1) + 3^(n-1) + 6^(n-1). - Álvar Ibeas, Mar 25 2015

Signature corrected and Ibeas formula adapted to the offset by Bruno Berselli, Mar 25 2015

cp's OEIS Frontend

A110143 Row sums of triangle A110141.

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A074528 a(n) = 2^n + 3^n + 6^n.

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A160450 Expansion of (1-34*x+276*x^2-584*x^3)/((1-3*x)*(1-4*x)*(1-8*x)*(1-24*x)).

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A152612 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 3.

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A160446 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 4.

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A160447 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 5.

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Sage

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A160448 Number of isomorphism classes of n-fold coverings of a connected graph with Betti number 6.

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A160450 Expansion of (1-34x+276x^2-584x^3)/((1-3x)(1-4x)(1-8x)(1-24x)).

A246985 Expansion of (1-8x+14x^2)/((1-2x)(1-3x)(1-6*x)).