A330200 -id:A330200 - cp's OEIS frontend

A330201 Expansion of e.g.f. Product_{k>=1} exp(-x^k) / (1 - x^k).

1, 0, 1, 2, 21, 44, 1045, 2694, 74473, 421784, 8776521, 52518410, 1843753021, 11476952772, 387068115421, 4277646186254, 125796357803985, 1343857519264304, 53205974734877713, 621203524858308114, 25357790175078682981, 388778926109137187420

Offset: 0

Ilya Gutkovskiy, Dec 05 2019

nonn

Cf. A000166, A000203, A051953, A053529, A293116, A330200.

nmax = 21; CoefficientList[Series[Product[Exp[-x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (n - 1)! Sum[(DivisorSigma[1, k] - k) a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 21}]

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000166.
E.g.f.: exp(Sum_{k>=1} (sigma(k) / k - 1) * x^k), where sigma = A000203.
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(cototient(k)/k), where cototient = A051953.
a(0) = 1; a(n) = (n - 1)! * Sum_{k=1..n} (sigma(k) - k) * a(n-k) / (n - k)!.
a(n) = Sum_{k=0..n} binomial(n,k) * A293116(k) * A053529(n-k).
a(n) ~ sqrt(-1/Pi + Pi/6) * n^(n - 1/2) / (2 * exp(n - 1/2 - sqrt(2*(-6 + Pi^2)*n/3))). - Vaclav Kotesovec, Aug 09 2021

A346964 Expansion of e.g.f. Product_{k>=1} exp(x^k) * (1 + x^k).

Original entry on oeis.org

1, 1, 2, 7, 40, 257, 2086, 19567, 207572, 2451745, 32226922, 462314711, 7178502112, 120315808417, 2157566463950, 41277697722367, 838883560646476, 18020304830796737, 408135672764386642, 9723868266912217255, 242827969365094823192, 6345340713682009241281

Offset: 0

Vaclav Kotesovec, Aug 09 2021

nonn

Cf. A000262, A088311, A330200.

nmax = 20; CoefficientList[Series[Product[Exp[x^k] * (1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[x/(1 - x)] * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k) / (1 - x^k)^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsQ[n-k],{k,0,n}], {n,0,20}]

a(n) = Sum_{k=0..n} binomial(n,k) * A000262(k) * A088311(n-k).
E.g.f.: Product_{k>=1} (1 + x^k) / (1 - x^k)^(A000010(k)/k).
a(n) ~ 2^(-3/2) * (4 + Pi^2/3)^(1/4) * exp(sqrt((4 + Pi^2/3)*n) - n - 1/2) * n^(n - 1/4).

cp's OEIS Frontend

A330201 Expansion of e.g.f. Product_{k>=1} exp(-x^k) / (1 - x^k).

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