A299019 Expansion of Product_{k>=1} (1 - x^k)^(k+1).

1, -2, -2, 2, 3, 6, -1, -2, -10, -14, -7, -2, 11, 26, 43, 30, 28, -6, -40, -92, -128, -132, -115, -48, 54, 200, 339, 484, 499, 476, 274, -32, -501, -998, -1539, -1924, -2042, -1838, -1139, 12, 1664, 3540, 5588, 7258, 8392, 8230, 6812, 3480, -1472, -8150, -15737, -23670, -30478

Offset: 0

Author

Ilya Gutkovskiy, Aug 11 2018

Keywords

• sign

Comments

Convolution of A010815 and A073592.

Convolution inverse of A005380.

Crossrefs

Cf. A005380, A010815, A073592, A092346.

Programs

Mathematica

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

Formula

G.f.: exp(-Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k).