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 1-8 of 8 results.

A343547 a(n) = n * Sum_{d|n} binomial(d+n-2,n-1)/d.

Original entry on oeis.org

1, 4, 9, 32, 75, 318, 931, 3712, 13014, 50110, 184767, 715656, 2704169, 10454976, 40126395, 155462016, 601080407, 2335849578, 9075135319, 35359120940, 137847221148, 538346579034, 2104098963743, 8234009441952, 32247603785500, 126414311404108, 495918587420145
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A332508, A343544, A343545, A343546.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 2, n-1]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-2, n-1)/d);

Formula

a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^n.
a(n) = [x^n] Sum_{k>=1} binomial(k+n-2,n-1) * x^k/(1 - x^k)^2.

A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

Original entry on oeis.org

1, 6, 13, 32, 40, 94, 91, 184, 204, 320, 297, 612, 468, 770, 850, 1184, 986, 1752, 1349, 2280, 2114, 2662, 2323, 4184, 3125, 4264, 4266, 5740, 4524, 7660, 5487, 8352, 7546, 9180, 8470, 13212, 9176, 12654, 12194, 16640, 12382, 19628, 14233, 20724, 19590, 22034, 18471, 30416, 21462
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.

Programs

  • Maple
    f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Apr 26 2021
  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.

A343549 a(n) = n * Sum_{d|n} binomial(d+n-1,n)/d.

Original entry on oeis.org

1, 5, 13, 49, 131, 545, 1723, 6809, 24484, 94445, 352727, 1366273, 5200313, 20135939, 77571083, 301034537, 1166803127, 4540794476, 17672631919, 68943346009, 269129827042, 1052178506615, 4116715363823, 16124644677569, 63205303337656, 247964681424725, 973469783435197
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A343544, A343545, A343546, A343547, A343548.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 1, n]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-1, n)/d);

Formula

a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^(n+1).
a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k)^2.
From Seiichi Manyama, Jun 14 2023: (Start)
a(n) = Sum_{d|n} binomial(d+n-1,d).
a(n) = [x^n] Sum_{k>=1} (1/(1 - x^k)^n - 1). (End)

A343545 a(n) = n * Sum_{d|n} binomial(d+3,4)/d.

Original entry on oeis.org

1, 7, 18, 49, 75, 177, 217, 428, 549, 890, 1012, 1824, 1833, 2849, 3360, 4732, 4862, 7506, 7334, 10810, 11382, 14729, 14973, 22188, 20850, 27482, 29052, 37408, 35989, 50490, 46407, 61824, 62106, 75854, 75390, 101673, 91427, 116033, 117624, 146680, 135792, 179886, 163228, 208208
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A073570, A309731, A343544, A343546, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 3, 4]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+3, 4)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^5 = Sum_{k>=1} binomial(k+3,4) * x^k/(1 - x^k)^2.

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

A343546 a(n) = n * Sum_{d|n} binomial(d+4,5)/d.

Original entry on oeis.org

1, 8, 24, 72, 131, 318, 469, 936, 1359, 2294, 3014, 5172, 6201, 9548, 12126, 17376, 20366, 29862, 33668, 47372, 54684, 71874, 80753, 111000, 119410, 154986, 173988, 220864, 237365, 309864, 324663, 411744, 445170, 542776, 578984, 731340, 749435, 918118, 981474
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A101289, A309731, A343544, A343545, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 4, 5]/# &]; Array[a, 40] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+4, 5)/d);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k)^2.

A366135 Expansion of Sum_{k>=1} k^3 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 11, 33, 98, 140, 366, 371, 820, 936, 1550, 1397, 3276, 2288, 4102, 4650, 6696, 5066, 10413, 7049, 13860, 12306, 15422, 12443, 27480, 17825, 25246, 25650, 36652, 24824, 51900, 30287, 54096, 46266, 55862, 52150, 93366, 51356, 77710, 75738, 116200, 69782, 137172
Offset: 1

Views

Author

Seiichi Manyama, Oct 28 2023

Keywords

Links

Crossrefs

Cf. A007437, A309731, A309732.
Cf. A001158, A328259.
Cf. A000203, A001157.

Programs

  • Mathematica
    a[n_] := (n * DivisorSigma[1, n] + DivisorSigma[2, n]) * n/2; Array[a, 50] (* Amiram Eldar, Dec 15 2023 *)
  • PARI
    a(n) = n*(n*sigma(n)+sigma(n, 2))/2;

Formula

a(n) = n * (n * sigma(n) + sigma_2(n))/2.
a(n) = Sum_{d|n} d^3 * binomial(n/d+1,2).
a(n) = Sum_{k=1..n} k*sigma_2(gcd(n,k)).
Sum_{k=1..n} a(k) ~ (Pi^2/48 + zeta(3)/8) * n^4. - Amiram Eldar, Dec 15 2023

A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -1, 9, -12, 20, -12, 35, -60, 72, -30, 77, -132, 104, -56, 210, -256, 170, -117, 209, -320, 378, -132, 299, -672, 425, -182, 594, -588, 464, -360, 527, -1040, 858, -306, 910, -1224, 740, -380, 1170, -1640, 902, -672, 989, -1364, 1890, -552, 1175, -2928, 1470, -775, 1938, -1872, 1484, -1080, 2090
Offset: 1

Views

Author

Seiichi Manyama, Jul 19 2023

Keywords

Links

Crossrefs

Cf. A320900, A364351.
Cf. A002129, A048272, A309731.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + 1)*(# + 1) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1+x^k)^3))

Formula

a(n) = (n/2) * Sum_{d|n} (-1)^(d+1) * (d+1) = (n/2) * (A002129(n) + A048272(n)).
Showing 1-8 of 8 results.

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