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.

A320900 Expansion of Sum_{k>=1} x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -2, 7, -12, 16, -17, 29, -48, 52, -42, 67, -105, 92, -79, 142, -184, 154, -143, 191, -262, 266, -189, 277, -441, 341, -262, 430, -495, 436, -402, 497, -712, 634, -444, 674, -897, 704, -553, 878, -1118, 862, -766, 947, -1189, 1222, -807, 1129, -1753, 1254, -992
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

Links

Crossrefs

Cf. A000203, A000217, A000593, A001157, A002129, A007437, A050999, A064027, A320901, A321543.
Cf. A363022, A363615, A363630.

Programs

  • Maple
    seq(coeff(series(add(x^k/(1+x^k)^3,k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    nmax = 50; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1)/2, {d, Divisors[n]}], {n, 50}]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)/2); \\ Amiram Eldar, Jan 04 2025

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*A000217(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)/2.
a(n) = A000593(n) + A050999(n) - (A000203(n) + A001157(n))/2.
a(n) = (A002129(n) + A321543(n)) / 2. - Amiram Eldar, Jan 04 2025

A363598 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.

Original entry on oeis.org

0, 1, -4, 11, -20, 32, -56, 95, -124, 146, -220, 328, -364, 400, -584, 775, -816, 881, -1140, 1486, -1600, 1552, -2024, 2712, -2620, 2562, -3400, 4064, -4060, 4112, -4960, 6231, -6208, 5730, -7216, 8947, -8436, 8000, -10248, 12230, -11480, 11232, -13244, 15752
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363613, A363614.
Cf. A002129, A138503, A363604.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^4)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+1, 3));

Formula

G.f.: Sum_{k>0} binomial(k+1,3) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+1,3) = (A002129(n) - A138503(n))/6.

A363613 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^5.

Original entry on oeis.org

0, 1, -5, 16, -35, 66, -126, 226, -335, 461, -715, 1082, -1365, 1695, -2420, 3286, -3876, 4581, -5985, 7791, -8986, 9912, -12650, 16242, -17585, 19111, -24086, 29115, -31465, 34106, -40920, 49662, -53080, 55030, -66206, 79412, -82251, 85406, -102640, 119931
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363598, A363614.
Cf. A363605.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^5)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+2, 4));

Formula

G.f.: Sum_{k>0} binomial(k+2,4) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+2,4).

A363614 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^6.

Original entry on oeis.org

0, 1, -6, 22, -56, 121, -252, 484, -798, 1232, -2002, 3145, -4368, 5937, -8630, 12112, -15504, 19678, -26334, 34902, -42762, 51129, -65780, 84337, -98336, 114388, -143304, 175869, -201376, 230120, -278256, 336744, -379000, 420394, -502250, 598459, -658008, 723065, -855042, 997962, -1086008
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363598, A363613.
Cf. A363606.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^#*Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^6)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+3, 5));

Formula

G.f.: Sum_{k>0} binomial(k+3,5) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+3,5).

A363617 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^4.

Original entry on oeis.org

0, 0, 1, -4, 10, -19, 35, -60, 85, -110, 165, -243, 286, -329, 466, -620, 680, -751, 969, -1254, 1366, -1375, 1771, -2323, 2310, -2314, 3010, -3609, 3654, -3734, 4495, -5580, 5622, -5304, 6590, -8115, 7770, -7467, 9426, -11190, 10660, -10498, 12341, -14623, 14740, -13409, 16215, -20179, 18459, -17410, 21506
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A363022, A363618.
Cf. A363607.

Programs

  • Mathematica
    a[n_] := -DivisorSum[n, (-1)^#*Binomial[#, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1+x^k)^4)))
  • PARI
    a(n) = -sumdiv(n, d, (-1)^d*binomial(d, 3));

Formula

G.f.: -Sum_{k>0} binomial(k,3) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d,3).

A363630 Expansion of Sum_{k>0} (1/(1+x^k)^3 - 1).

Original entry on oeis.org

-3, 3, -13, 18, -24, 21, -39, 63, -68, 48, -81, 127, -108, 87, -170, 216, -174, 156, -213, 294, -302, 201, -303, 497, -375, 276, -474, 537, -468, 426, -531, 777, -686, 462, -726, 965, -744, 573, -938, 1200, -906, 798, -993, 1251, -1306, 831, -1179, 1875, -1314, 1023, -1562, 1722, -1488, 1290, -1698
Offset: 1

Views

Author

Seiichi Manyama, Jun 12 2023

Keywords

Links

Crossrefs

Cf. A002129, A048272, A321543, A363629, A363631.
Cf. A320900, A363022, A363615.

Programs

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

Formula

G.f.: Sum_{k>0} binomial(k+2,2) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+2,2).
a(n) = -(A321543(n) + 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A363618 Expansion of Sum_{k>0} x^(4*k)/(1+x^k)^5.

Original entry on oeis.org

0, 0, 0, 1, -5, 15, -35, 71, -126, 205, -330, 511, -715, 966, -1370, 1891, -2380, 2949, -3876, 5051, -6020, 6985, -8855, 11207, -12655, 14235, -17676, 21442, -23751, 26260, -31465, 37851, -41250, 43996, -52400, 62350, -66045, 69939, -82966, 96511, -101270
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A363022, A363617.
Cf. A363608.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 4] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^5)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d, 4));

Formula

G.f.: Sum_{k>0} binomial(k,4) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d,4).
Showing 1-7 of 7 results.

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