A319109 Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).

1, 0, -1, -2, -2, -2, 0, 2, 7, 8, 12, 10, 9, -2, -10, -32, -40, -62, -62, -70, -37, -20, 57, 106, 224, 272, 388, 376, 431, 272, 192, -184, -414, -1012, -1321, -2020, -2157, -2700, -2318, -2352, -1014, -272, 2280, 3798, 7464, 9200, 13257, 13958, 17098, 14846, 15266

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• sign

Comments

Convolution of A000009 and A255528.

Convolution inverse of A052812.

Crossrefs

Cf. A000009, A052812, A255528, A305628.

Programs

Maple

a:=series(mul(1/(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/(1 + x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^k x^(2 k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

Formula

G.f.: exp(Sum_{k>=1} (-1)^k*x^(2*k)/(k*(1 - x^k)^2)).