A318844 Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

1, 0, 1, 1, 2, 2, 5, 4, 8, 10, 15, 17, 29, 31, 48, 60, 81, 99, 143, 167, 231, 287, 374, 460, 615, 740, 964, 1194, 1512, 1856, 2379, 2877, 3635, 4460, 5540, 6759, 8433, 10192, 12608, 15335, 18774, 22726, 27868, 33525, 40863, 49292, 59652, 71694, 86780, 103818, 125118, 149778, 179608

Offset: 0

Author

Ilya Gutkovskiy, Sep 04 2018

Keywords

• nonn

Comments

Convolution of A081362 and A107742.

Weigh transform of A032741.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000005, A000203, A032741, A081362, A107742, A318783.

Programs

Maple

with(numtheory): a:=series(mul((1+x^k)^(tau(k)-1),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 52; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 1) x^k/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

Formula

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

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