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 1-5 of 5 results.

A363642 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^3.

Original entry on oeis.org

1, 4, 7, 17, 16, 55, 29, 129, 100, 311, 67, 1135, 92, 1919, 1486, 5409, 154, 17038, 191, 33491, 20938, 67871, 277, 262861, 9701, 373127, 296110, 978727, 436, 3134821, 497, 5051969, 3898522, 10027655, 474146, 39352069, 704, 49808159, 48362926, 127403221, 862, 411286429, 947
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363643, A363644.
Cf. A167531, A363645.
Cf. A007437, A363650.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+1, 2));

Formula

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

A363651 Expansion of Sum_{k>0} x^(2*k)/(1 - (k*x)^k)^3.

Original entry on oeis.org

0, 1, 3, 7, 10, 28, 21, 125, 117, 686, 55, 9049, 78, 21596, 206310, 508025, 136, 8701561, 171, 229315221, 303797886, 14418152, 253, 88452515089, 305175781550, 327156038, 377734977126, 27160609347425, 406, 2458857416866336, 465, 9570181420417521
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363650, A363652.
Cf. A363643.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-2*n/#) * Binomial[#, 2] &]; Array[a, 33] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-2*n/d)*binomial(d, 2));

Formula

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

A363652 Expansion of Sum_{k>0} x^(3*k)/(1 - (k*x)^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 33, 29, 132, 45, 777, 66, 3918, 4466, 22377, 120, 311655, 153, 992586, 7971806, 2949330, 231, 483657349, 58594026, 69206316, 10847774018, 64754136132, 378, 696335917637, 435, 23096840946129, 12709329142142, 32212255248, 1434580813047030
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363650, A363651.
Cf. A363644.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n - 3*n/#)*Binomial[# - 1, 2] &]; Array[a, 35] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-3*n/d)*binomial(d-1, 2));

Formula

a(n) = Sum_{d|n} (n/d)^(n-3*n/d) * binomial(d-1,2).

A363664 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 56, 127, 1100, 1717, 19300, 64406, 383010, 352717, 23214660, 5200301, 191172406, 3465549077, 20859527460, 1166803111, 1010698826825, 17672631901, 102589250081802, 286539905316908, 75260204476154, 4116715363801, 548610025890719156
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A343548, A363662, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 1, n] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^(n+1).

A363667 a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 37, 71, 751, 925, 13161, 45676, 262911, 184757, 18014557, 2704157, 133062875, 2838201061, 16907954129, 601080391, 830283170617, 9075135301, 87074953375981, 246003195539410, 53321730394923, 2104098963721, 479275771000215865, 1952680410445479976
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A342628, A359112, A363650.
Cf. A363664, A363666.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - (k*x)^k)^n.
Showing 1-5 of 5 results.

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