A288415 Expansion of Product_{k>=1} (1 + x^k)^(sigma_3(k)).

1, 1, 9, 37, 137, 487, 1749, 5901, 19695, 63832, 202905, 632689, 1941394, 5860868, 17448558, 51255292, 148726841, 426605755, 1210569740, 3400427281, 9460683203, 26083933370, 71300381025, 193313191005, 520057831035, 1388722752205, 3682100198763

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..5402

Formula

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.

a(n) ~ exp(5*Pi^(4/5) * Zeta(5)^(1/5) * n^(4/5) / 2^(12/5)) * Zeta(5)^(1/10) / (2^(169/240) * sqrt(5) * Pi^(1/10) * n^(3/5)). - Vaclav Kotesovec, Mar 23 2018

G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^3). - Ilya Gutkovskiy, Aug 26 2018

Maple

with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[3](k)), k=1..n), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

Prog

(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k, 3))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Crossrefs

Cf. A288391, A288392, A288420.

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), A288414 (m=2), this sequence (m=3).

Sequence in context: A213570 A271908 A257448 * A026620 A048878 A246315

Adjacent sequences: A288412 A288413 A288414 * A288416 A288417 A288418

Keyword

nonn

Author

Seiichi Manyama, Jun 09 2017

Status

approved