A301554 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

1, 2, 6, 14, 32, 66, 138, 266, 512, 948, 1730, 3074, 5408, 9306, 15854, 26594, 44150, 72378, 117620, 189074, 301516, 476518, 747514, 1163470, 1798920, 2762040, 4215194, 6393196, 9642596, 14462518, 21581386, 32040562, 47345342, 69635866, 101974722, 148692638

Offset

0,2

Comments

Convolution of A006171 and A107742.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))/(1 - x^(i*j)). - Ilya Gutkovskiy, May 23 2018

Conjecture: log(a(n)) ~ Pi * sqrt(n*log(n)/2). - Vaclav Kotesovec, Sep 03 2018

Maple

with(numtheory): seq(coeff(series(mul(((1+x^k)/(1-x^k))^sigma[0](k), k=1..n), x, n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 29 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, prod(j=1, m+2, (1+x^(j*k))/(1-x^(j*k)) ))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(j*k))/(1-x^(j*k)): j in [1..(m+2)]]): k in [1..(m+2)]]))); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000005, A006171, A107742, A320237.

Sequence in context: A327049 A035592 A327050 * A217941 A346679 A232434

Adjacent sequences: A301551 A301552 A301553 * A301555 A301556 A301557

Keyword

nonn

Author

Vaclav Kotesovec, Mar 23 2018

Status

approved