A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

1, -1, -1, -1, 1, 1, 0, 2, 2, -1, -2, 0, -1, -1, -4, -1, 2, 0, -1, 2, 2, 5, -1, 4, 8, -4, -5, 0, -1, -1, -6, -1, 3, -7, -9, -5, 1, 3, -3, 3, 17, 0, -6, 8, 12, 8, 0, 8, 17, -11, -9, -10, 0, -2, -20, 5, 14, -18, -25, -10, 1, -7, -21, 2, 29, -12, -17, 6, 17, 32, -4

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

Convolution inverse of A107742.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 26 2018

Maple

with(numtheory): seq(coeff(series(exp(-add(sigma(k)*x^k/(k*(1-x^(2*k))), k=1..n)), x, n+1), x, n), n = 0 .. 70); # Muniru A Asiru, Jan 30 2019

Mathematica

A109386[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2] &] &]; a[0] = 1; a[n_] := a[n] = -(1/n) Sum[A109386[k] a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 04 2017 *)
CoefficientList[Series[1/Product[Product[1+x^(j*k), {j, 1, 100}], {k, 1, 100}], {x, 0, 80}], x] (* G. C. Greubel, Oct 29 2018 *)

Prog

(PARI) m=80; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, 2*m, 1+x^(j*k) )))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[(&*[1 + x^(j*k): j in [1..2*m]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A107742, A109386.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): this sequence (m=0), A288421 (m=1), A288422 (m=2), A288423 (m=3).

Sequence in context: A074942 A043754 A144191 * A145783 A145785 A094022

Adjacent sequences: A288004 A288005 A288006 * A288008 A288009 A288010

Keyword

sign

Author

Seiichi Manyama, Jun 04 2017

Status

approved