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A110143
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Row sums of triangle A110141.
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19
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1, 1, 4, 11, 43, 161, 901, 5579, 43206, 378360, 3742738, 40853520, 488029621, 6323154547, 88308425755, 1322120265238, 21122364398761, 358647945023885, 6449299885654827, 122436442904193940, 2447046870232798369, 51358050784584629338, 1129314001779283063606
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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]
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(Sage)
return sum(p.aut() for p in Partitions(n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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