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A182269
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Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.
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20
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1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 65, 88, 132, 177, 253, 340, 478, 633, 874, 1150, 1562, 2045, 2736, 3553, 4713, 6082, 7969, 10234, 13301, 16973, 21889, 27789, 35577, 44961, 57179, 71906, 90950, 113874, 143204, 178592, 223505, 277599, 345822, 427934, 530797
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: Product_{k>0} 1/(1-x^k)^A038548(k).
G.f.: Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))). - Vaclav Kotesovec, Aug 19 2019
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EXAMPLE
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a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 2: 2 = 1*1 + 1*1 = 1*2.
a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.
a(4) = 6: 4 = 1*1 + 1*1 + 1*1 + 1*1 = 1*1 + 1*1 + 1*2 = 1*1 + 1*3 = 1*2 + 1*2 = 2*2 = 1*4.
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*ceil(tau(d)/2), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60);
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Ceiling[DivisorSigma[0, d]/2], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[1/(1 - x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)
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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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