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A110143 Row sums of triangle A110141.

%I A110143 #81 Jul 03 2022 06:45:29
%S A110143 1,1,4,11,43,161,901,5579,43206,378360,3742738,40853520,488029621,
%T A110143 6323154547,88308425755,1322120265238,21122364398761,358647945023885,
%U A110143 6449299885654827,122436442904193940,2447046870232798369,51358050784584629338,1129314001779283063606
%N A110143 Row sums of triangle A110141.
%C A110143 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.
%C A110143 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
%C A110143 Number of isomorphism classes of n-fold coverings of a connected graph with circuit rank 2 [Kwak and Lee]. - _Álvar Ibeas_, Mar 25 2015
%D A110143 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.
%D A110143 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]
%H A110143 Alois P. Heinz, <a href="/A110143/b110143.txt">Table of n, a(n) for n = 0..450</a>
%H A110143 Mathieu Bogaerts and Peter Dukes, <a href="http://dx.doi.org/10.1016/j.disc.2014.03.002">Semidefinite programming for permutation codes</a>, Discrete Math. 326 (2014), 34--43. MR3188985.
%H A110143 Nicholas Dub, <a href="https://tel.archives-ouvertes.fr/tel-03641958/">Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2</a>, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
%H A110143 J. B. Geloun, S. Ramgoolam, <a href="http://arxiv.org/abs/1307.6490">Counting Tensor Model Observables and Branched Covers of the 2-Sphere</a>, arXiv:1307.6490 [hep-th], 2013.
%H A110143 Joseph Ben Geloun, Sanjaye Ramgoolam, <a href="https://arxiv.org/abs/2106.01470">All-orders asymptotics of tensor model observables from symmetries of restricted partitions</a>, arXiv:2106.01470 [hep-th], Jun 02 2021.
%H A110143 J. H. Kwak and J. Lee, <a href="http://dx.doi.org/10.4153/CJM-1990-039-3">Isomorphism classes of graph bundles</a>. Can. J. Math., 42(4), 1990, pp. 747-761.
%H A110143 MathOverflow, <a href="http://mathoverflow.net/questions/41337/a-general-formula-for-the-number-of-conjugacy-classes-of-mathbbs-n-times-m/">A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n</a> [From Peter Dukes, May 12 2014]
%H A110143 Igor Pak, Greta Panova, Damir Yeliussizov, <a href="https://arxiv.org/abs/1804.04693">On the largest Kronecker and Littlewood-Richardson coefficients</a>, arXiv:1804.04693 [math.CO], 2018.
%F A110143 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
%F A110143 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
%p A110143 # Using a function from _Alois P. Heinz_ in A279038:
%p A110143 b:= proc(n, i) option remember; `if`(n=0, [1],
%p A110143       `if`(i<1, [], [seq(map(x-> x*i^j*j!,
%p A110143       b(n-i*j, i-1))[], j=0..n/i)]))
%p A110143     end:
%p A110143 seq(add(i, i=b(n$2)), n=0..22); # _Peter Luschny_, Dec 19 2016
%t A110143 Table[Total[Apply[Times, Tally[#]/.{a_Integer,b_}->a^b b!]& /@ IntegerPartitions[n]],{n,0,21}] (* _Wouter Meeussen_, Oct 17 2014 *)
%t A110143 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_ *)
%t A110143 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 *)
%t A110143 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 *)
%o A110143 (Sage)
%o A110143 def A110143(n):
%o A110143     return sum(p.aut() for p in Partitions(n))
%o A110143 [A110143(n) for n in range(9)]
%o A110143 # _Álvar Ibeas_, Mar 26 2015
%Y A110143 Cf. A110141, A000041, A057005, A096161, A241584, A241585, A279819.
%Y A110143 Third column of A160449.
%K A110143 nonn
%O A110143 0,3
%A A110143 _Paul D. Hanna_, Jul 14 2005