A318579 Expansion of Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^(i*j).

1, 2, 10, 30, 98, 270, 786, 2046, 5418, 13556, 33726, 81002, 192902, 447562, 1027990, 2316750, 5165398, 11345298, 24668952, 52972902, 112688802, 237193354, 494933514, 1023238806, 2098662698, 4269141516, 8620916966, 17280687472, 34405835066, 68044209950, 133732805458

Offset

0,2

Comments

Convolution of A280540 and A280541.

Links

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

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*tau(k)), where tau(k) = number of divisors of k (A000005).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (1 - (-1)^(k/d))*d^2*tau(d) ) * x^k/k).

log(a(n)) ~ 3^(2/3) * (7*Zeta(3))^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Sep 03 2018

Maple

a:=series(mul(mul(((1+x^(i*j))/(1-x^(i*j)))^(i*j), j=1..100), i=1..100), x=0, 31): seq(coeff(a, x, n), n=0..30); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 30; CoefficientList[Series[Product[Product[((1 + x^(i j))/(1 - x^(i j)))^(i j), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k DivisorSigma[0, k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(1 - (-1)^(k/d)) d^2 DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 30}]

Crossrefs

Cf. A000005, A038040, A156616, A280540, A280541, A301554, A301555.

Sequence in context: A355775 A234471 A196317 * A064932 A343513 A285956

Adjacent sequences: A318576 A318577 A318578 * A318580 A318581 A318582

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 29 2018

Status

approved