A327927 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j=1..i} x^(i*j) / (i*j)).

1, 1, 2, 6, 30, 150, 1020, 7140, 63420, 611100, 6625080, 72875880, 977213160, 12876743880, 190951160400, 2975661889200, 51767677962000, 886225654314000, 17136230971860000, 329530590793404000, 7035395004749311200, 151961029211943151200

Offset: 0

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

• nonn

Crossrefs

Cf. A000005, A028260, A028342, A038548, A206303, A327940.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Exp[Sum[Ceiling[DivisorSigma[0, k]/2] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Ceiling[DivisorSigma[0, k]/2] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 20; CoefficientList[Series[Exp[Sum[-(x^(k*(1 + k))*LerchPhi[x^k, 1, 1 + k] + Log[1 - x^k])/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 06 2019 *)

Formula

E.g.f.: exp(Sum_{k>=1} ceiling(A000005(k)/2) * x^k / k).

E.g.f.: exp(Sum_{k>=1} A038548(k) * x^k / k).

E.g.f.: Product_{k>=1} 1 / (1 - x^A028260(k))^(1/A028260(k)).