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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000
Crossrefs
Programs
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Magma
m:=80; R
:=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 -
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
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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 *)
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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
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