A319108 Expansion of Product_{k>=1} (1 - x^k)^(k-1).

1, 0, -1, -2, -3, -2, -1, 4, 8, 14, 17, 18, 9, -4, -27, -58, -88, -114, -122, -106, -48, 48, 200, 376, 577, 746, 862, 840, 646, 208, -486, -1450, -2622, -3888, -5086, -5950, -6204, -5492, -3547, -44, 5036, 11732, 19582, 28034, 35932, 42042, 44519, 41660, 31450, 12382, -16721

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• sign

Comments

Convolution of A000041 and A073592.

Convolution inverse of A052847.

Crossrefs

Cf. A000041, A000203, A001157, A052847, A073592, A299019.

Programs

Maple

a:=series(mul((1-x^k)^(k-1),k=1..100),x=0,51): seq(coeff(a,x,n),n=0..50); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; 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 (1 - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

Formula

G.f.: exp(Sum_{k>=1} (sigma_1(k) - sigma_2(k))*x^k/k), where sigma_1(k) = sum of divisors of k (A000203) and sigma_2(k) = sum of squares of divisors of k (A001157).