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 1-4 of 4 results.

A356039 a(n) = Sum_{k=1..n} binomial(n,k) * sigma_3(k).

Original entry on oeis.org

1, 11, 58, 243, 866, 2804, 8485, 24387, 67333, 180086, 469338, 1196976, 2996956, 7385837, 17954243, 43125267, 102494548, 241309031, 563341508, 1305142418, 3002938045, 6866090880, 15609292379, 35299794600, 79443050541, 177989130174, 397124963671, 882642816697, 1954708794400
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 24 2022

Keywords

Comments

For m>0, Sum_{k=1..n} binomial(n,k) * sigma_m(k) ~ zeta(m+1) * n^m * 2^(n-m).

Links

Crossrefs

Cf. A001158, A160399, A185003, A356038.
Cf. A006218, A024916, A064602, A064603.

Programs

  • Maple
    with(numtheory): seq(add(sigma[3](i)*binomial(n,i), i=1..n), n=1..60); # Ridouane Oudra, Oct 31 2022
  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[3, k], {k, 1, n}], {n, 1, 40}]
  • PARI
    a(n) = sum(k=1, n, binomial(n,k) * sigma(k, 3)); \\ Michel Marcus, Jul 24 2022

Formula

a(n) ~ Pi^4 * n^3 * 2^(n-4) / 45.
a(n) = Sum_{i=1..n} Sum_{j=1..n} (i^3)*binomial(n,i*j). - Ridouane Oudra, Oct 31 2022

A356339 a(n) = Sum_{k=1..n} binomial(2*n, n-k) * sigma_2(k).

Original entry on oeis.org

1, 9, 55, 297, 1496, 7215, 33783, 154825, 698077, 3107424, 13690161, 59802471, 259377080, 1118176887, 4795381640, 20472223529, 87051685546, 368857919085, 1558036408998, 6562564601592, 27571934249754, 115574440020477, 483444570596465, 2018365519396135, 8411811012694246
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 04 2022

Keywords

Crossrefs

Cf. A001157, A064602, A351146, A356038.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n, n-k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
  • PARI
    a(n) = sum(k=1, n, binomial(2*n, n-k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022

Formula

a(n) ~ zeta(3) * n * 4^(n-1).

A356342 a(n) = Sum_{k=1..n} binomial(2*n, k) * sigma_2(k).

Original entry on oeis.org

2, 34, 281, 2178, 12397, 79729, 398932, 2224354, 10959221, 56341309, 255685080, 1334248401, 5892916876, 28082515768, 127714609741, 604178948098, 2590365128017, 12284868071365, 52160408294826, 241445420212893, 1049251819301974, 4674022621994716, 19563451165603647
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 04 2022

Keywords

Crossrefs

Cf. A001157, A064602, A351146, A356038.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n, k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
  • PARI
    a(n) = sum(k=1, n, binomial(2*n, k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022

Formula

a(n) ~ zeta(3) * n^2 * 2^(2*n-1).

A356345 a(n) = Sum_{k=1..n} binomial(2*k, k) * sigma_2(k).

Original entry on oeis.org

2, 32, 232, 1702, 8254, 54454, 226054, 1320004, 5744424, 29762704, 115825408, 683698168, 2451800168, 12480950168, 52811505368, 257779918358, 934525722158, 5063712283658, 17858697779258, 93122902514978, 362251839734978, 1645752207604178, 6009470493232178, 33419933623867178
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 04 2022

Keywords

Comments

The average value of a(n) is zeta(3) * n^(3/2) * 4^(n+1) / (3*sqrt(Pi)).

Crossrefs

Cf. A001157, A064602, A351146, A356038.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k, k]*DivisorSigma[2, k], {k, 1, n}], {n, 1, 30}]
  • PARI
    a(n) = sum(k=1, n, binomial(2*k, k) * sigma(k, 2)); \\ Michel Marcus, Aug 05 2022
Showing 1-4 of 4 results.

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