A319359 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k)).

1, -1, -3, 0, 0, 9, 5, 4, -1, -27, -2, -33, -17, 8, 43, 92, 36, 100, -8, -11, -136, -120, -296, -363, -13, -203, 286, 306, 1010, 667, 724, 790, 151, -258, -1207, -964, -3325, -2059, -2924, -1992, -2116, 1277, 3625, 4437, 7724, 7734, 11524, 5801, 9685, -855, -2799, -13409, -16423

Offset: 0

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

• sign

Links

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

Crossrefs

Cf. A000005, A000203, A007425, A174465, A174466, A280473, A288098.

Programs

Maple

a:=series(mul(mul(mul(1-x^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

Mathematica

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

Formula

G.f.: Product_{k>=1} (1 - x^k)^A007425(k).

G.f.: exp(-Sum_{k>=1} A174466(k)*x^k/k).