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A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

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

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}]

PROG

(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

CROSSREFS

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

Sequence in context: A082674 A292773 A228338 * A253951 A102172 A011983

Adjacent sequences:  A309728 A309729 A309730 * A309732 A309733 A309734

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 14 2019

STATUS

approved

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Last modified July 14 18:26 EDT 2020. Contains 335729 sequences. (Running on oeis4.)