cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

Comments

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

Links

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

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

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

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

A163920 Expansion of Sum_{k>0} k*(k+1)/2 * x^k / (1 - (-x)^k)^3.

Original entry on oeis.org

1, 0, 12, 9, 30, 0, 56, 60, 126, 0, 132, 126, 182, 0, 420, 316, 306, 0, 380, 330, 798, 0, 552, 888, 875, 0, 1296, 630, 870, 0, 992, 1536, 1914, 0, 2100, 1467, 1406, 0, 2652, 2360, 1722, 0, 1892, 1518, 4860, 0, 2256, 4872, 3234, 0, 4488, 2106, 2862, 0, 5060
Offset: 1

Views

Author

Paul D. Hanna, Aug 06 2009

Keywords

Links

Crossrefs

Cf. A143520 (variant), A034715.

Programs

  • Mathematica
    CoefficientList[Series[Sum[((k(k+1))/2 x^k)/(1-(-x)^k)^3,{k,100}],{x,0,100}],x] (* Harvey P. Dale, May 08 2021 *)
  • PARI
    {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k*(k+1)/2 * x^k / (1 - (-x)^k)^3, x*O(x^n)), n))}

Formula

a(4n+2) = 0.
a(2n+1) = A034715(2n+1), where A034715 is the Dirichlet convolution of triangular numbers with themselves.
a(n) = (n/4) * Sum_{d|n} (-1)^(n+d) * (d+1) * (n/d+1). - Seiichi Manyama, Jul 17 2023
Showing 1-3 of 3 results.

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