A321241 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))/(1 + x^(i*j*k)).

1, -2, -4, 4, 8, 16, -12, -28, -28, -56, 64, 68, 152, 144, -20, -72, -678, -508, -424, 92, 824, 1512, 2204, 1036, 936, -1900, -2936, -6444, -5656, -4384, -4808, 6540, 10080, 21256, 20296, 24424, 13520, -7856, -28836, -55744, -72240, -92960, -48424, -40772, 36168, 106464, 199996

Offset

0,2

Comments

Convolution inverse of A305050.

Links

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

Formula

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

G.f.: Product_{k>=1} theta_4(x^k)^tau(k), where theta_4() is the Jacobi theta function and tau() is the number of divisors. - Ilya Gutkovskiy, May 18 2019

Mathematica

With[{nmax=50}, CoefficientList[Series[Product[(1 - x^(i*j*k))/(1 + x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]] (* G. C. Greubel, Nov 01 2018 *)

Prog

(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, ((1-x^k)/(1+x^k))^sumdiv(k, x, sumdiv(x, y, 1 )))) \\ G. C. Greubel, Nov 01 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(   (&*[(&*[(&*[(1 - x^(i*j*k))/(1 + x^(i*j*k)): i in [1..m]]): j in [1..m]]): k in [1..m]]))); // G. C. Greubel, Nov 01 2018

Crossrefs

Cf. A000005, A002448, A007425, A305050, A320908.

Sequence in context: A181217 A223569 A227594 * A290128 A291778 A089887

Adjacent sequences: A321238 A321239 A321240 * A321242 A321243 A321244

Keyword

sign

Author

Seiichi Manyama, Nov 01 2018

Status

approved