A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

1, 5, 9, 20, 20, 48, 35, 76, 72, 110, 77, 204, 104, 196, 210, 288, 170, 405, 209, 480, 378, 440, 299, 816, 425, 598, 594, 868, 464, 1200, 527, 1104, 858, 986, 910, 1800, 740, 1216, 1170, 1960, 902, 2184, 989, 1980, 1890, 1748, 1175, 3216, 1470, 2475, 1938, 2704, 1484, 3456, 2090

Offset: 1

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

• nonn

Comments

Dirichlet convolution of natural numbers (A000027) with triangular numbers (A000217).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000027, A000203, A000217, A007437, A007503, A034715, A038040, A064987, A309732.

Programs

Maple

with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # Ridouane Oudra, Nov 28 2019

Mathematica

nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]

PARI

a(n)=sumdiv(n,d,binomial(n/d+1,2)*d); \\ Andrew Howroyd, Aug 14 2019

PARI

a(n)=n*(numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Apr 19 2021

Formula

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.

a(n) = n * (d(n) + sigma(n))/2.

Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.

a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - Ridouane Oudra, Nov 28 2019