A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

1, 7, 15, 38, 40, 108, 77, 188, 180, 290, 187, 600, 260, 560, 630, 888, 442, 1323, 551, 1620, 1218, 1364, 805, 3024, 1325, 1898, 1998, 3136, 1276, 4680, 1457, 4080, 2970, 3230, 3290, 7470, 2072, 4028, 4134, 8200, 2542, 9072, 2795, 7656, 7830, 5888, 3337, 14496, 4998, 9825, 7038

Offset: 1

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

• nonn

Comments

Dirichlet convolution of triangular numbers (A000217) with squares (A000290).

a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.

Programs

Magma

[n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // Marius A. Burtea, Nov 29 2019

Maple

with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # Ridouane Oudra, Nov 28 2019

Mathematica

nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]

PARI

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

PARI

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

Formula

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.

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

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

a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - Omar E. Pol, Aug 17 2019

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