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A305049
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Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).
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0
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1, 1, 3, 8, 27, 67, 216, 569, 1747, 4812, 14041, 39483, 115408, 326385, 941735, 2684170, 7725097, 22063737, 63354066, 181223899, 519883185, 1488316952, 4266788191, 12219763777, 35023995792, 100326757107, 287503501905, 823654031283, 2360146144917, 6761847714698, 19374935267810
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).
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MAPLE
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A:= proc(n, k) option remember; `if`(k=1, 1,
add(A(d, k-1), d=numtheory[divisors](n)))
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(A(j$2)*a(n-j), j=1..n))
end:
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MATHEMATICA
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nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]
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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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