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 1-7 of 7 results.

A363639 Expansion of Sum_{k>0} (1/(1 - k*x^k)^3 - 1).

Original entry on oeis.org

3, 12, 19, 51, 36, 180, 57, 405, 352, 918, 111, 3990, 144, 5064, 6534, 15945, 222, 51462, 267, 99354, 82478, 160812, 369, 808490, 66051, 861630, 1090342, 2593614, 552, 8966414, 621, 13039761, 13831470, 22415778, 3166218, 114011229, 852, 110103540, 167426822
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A362683, A363640.
Cf. A363642, A363643, A363644.
Cf. A363628, A363647.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + 2, 2] &]; Array[a, 40] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+2, 2));

Formula

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

A363650 Expansion of Sum_{k>0} x^k/(1 - (k*x)^k)^3.

Original entry on oeis.org

1, 4, 7, 23, 16, 199, 29, 1445, 4420, 13271, 67, 751597, 92, 2585423, 66565486, 218693769, 154, 14527231822, 191, 399614708821, 4080186211018, 856004218103, 277, 2754664372347481, 1430511474609701, 908626846503767, 900580521111136750, 5626675967703843613, 436
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363651, A363652.
Cf. A007437, A363642.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + 1, 2] &]; Array[a, 30] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+1, 2));

Formula

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

A363643 Expansion of Sum_{k>0} x^(2*k)/(1 - k*x^k)^3.

Original entry on oeis.org

0, 1, 3, 7, 10, 22, 21, 53, 45, 126, 55, 373, 78, 764, 390, 2009, 136, 5995, 171, 12501, 5334, 28392, 253, 92353, 1550, 160070, 79110, 394913, 406, 1167406, 465, 2081649, 1083126, 4457010, 69650, 14534659, 666, 22414016, 13818246, 53200481, 820, 158661826, 903
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363642, A363644.
Cf. A363651.

Programs

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

Formula

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

A363644 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, 27, 29, 60, 45, 145, 66, 318, 146, 789, 120, 2199, 153, 4890, 1406, 11730, 231, 34175, 426, 67884, 20738, 163956, 378, 453341, 435, 882153, 295742, 1966608, 10230, 5656484, 630, 10027674, 3897938, 23069916, 780, 63648517, 861, 113050536
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363642, A363643.
Cf. A363610, A363652.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(# - 3)*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-3)*binomial(d-1, 2));

Formula

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

A363645 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^4.

Original entry on oeis.org

1, 5, 11, 29, 36, 109, 85, 297, 256, 801, 287, 2881, 456, 5965, 3766, 17489, 970, 57385, 1331, 125681, 63498, 294933, 2301, 1072865, 24801, 1867009, 1087030, 4942561, 4496, 15697761, 5457, 28721057, 16895770, 63511593, 1404306, 225177013, 9140, 348661477
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A167531, A363642.
Cf. A059358.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(# - 1)*Binomial[# + 2, 3] &]; Array[a, 40] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+2, 3));

Formula

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

A363663 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 46, 127, 596, 1717, 7792, 24806, 108450, 352717, 1563914, 5200301, 22539046, 77876117, 331982444, 1166803111, 4945693769, 17672631901, 74053888812, 269344740908, 1118110015874, 4116715363801, 16984153623296, 63205318063252, 259049084680612
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A343548, A363661, A363664.

Programs

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

Formula

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

A363666 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 29, 71, 355, 925, 4425, 13276, 60111, 184757, 856357, 2704157, 12137147, 40367461, 176999505, 601080391, 2616894901, 9075135301, 38884056181, 138014377810, 583674491643, 2104098963721, 8823912454489, 32247616479976, 133998376789707
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A332508, A363663, A363667.

Programs

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

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^n.
Showing 1-7 of 7 results.

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