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A304965
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Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).
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3
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1, 1, 3, 6, 19, 30, 96, 152, 461, 775, 1883, 3271, 8751, 14370, 34004, 59491, 140450, 239746, 541817, 932681, 2089189, 3606641, 7719178, 13398411, 28848808, 49603982, 103047935, 179154858, 370200348, 639269735, 1295389370, 2241994088, 4511677298, 7798101800, 15408901600
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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.: Product_{k>=1} 1/(1 - x^k)^A163767(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(add(d*
A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]
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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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