A318783 Expansion of Product_{k>=1} 1/(1 - x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

1, 0, 1, 1, 3, 2, 7, 5, 14, 13, 27, 26, 57, 53, 102, 110, 192, 204, 353, 381, 626, 704, 1094, 1246, 1920, 2185, 3252, 3800, 5503, 6440, 9213, 10827, 15194, 18035, 24836, 29579, 40369, 48103, 64758, 77635, 103279, 123882, 163506, 196286, 256688, 308836, 400329, 481847, 620832

Offset: 0

Author

Ilya Gutkovskiy, Sep 03 2018

Keywords

• nonn

Comments

Convolution of A010815 and A006171.

Euler transform of A032741.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000005, A000203, A006171, A010815, A032741, A318784.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      (tau(d)-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018

Mathematica

nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A032741(k).

G.f.: exp(Sum_{k>=1} sigma_1(k)*x^(2*k)/(k*(1 - x^k))), where sigma_1(k) = sum of divisors of k (A000203).