A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).

1, 1, 5, 15, 41, 107, 286, 700, 1735, 4162, 9803, 22673, 51822, 116376, 258548, 567197, 1230763, 2642958, 5622616, 11850537, 24769248, 51353095, 105662389, 215838649, 437890022, 882562763, 1767741732, 3519599996, 6967592060, 13717874719, 26865949075

Offset: 0

Author

Seiichi Manyama, Jun 08 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A275585, A288389, A288419.

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), this sequence (m=2), A288415 (m=3).

Programs

Magma

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(2,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Maple

with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[2](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[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

PARI

m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

Formula

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.

a(n) ~ exp(2^(5/4) * (7*Zeta(3))^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (2^(13/4) * 3^(7/4) * (7*Zeta(3))^(1/4))) * (7*Zeta(3))^(1/8) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018

G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^2). - Ilya Gutkovskiy, Aug 26 2018