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

Author

Paul D. Hanna, Jul 14 2005

Keywords

• nonn

Comments

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

References

• 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]

Links

• 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.

Crossrefs

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

Third column of A160449.

Programs

Maple

# 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

Mathematica

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 *)

Sage

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

Formula

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