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.

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

Original entry on oeis.org

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363598, A363613, A363614.
Cf. A002129, A069153, A321543.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d, 2));

Formula

G.f.: Sum_{k>0} binomial(k,2) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.

A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

Original entry on oeis.org

1, -3, 11, -23, 36, -49, 85, -143, 176, -188, 287, -433, 456, -479, 726, -959, 970, -1024, 1331, -1748, 1866, -1741, 2301, -3153, 2961, -2824, 3830, -4559, 4496, -4514, 5457, -6943, 6842, -6174, 7890, -9844, 9140, -8553, 11126, -13348, 12342, -11998, 14191, -16941
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

Links

Crossrefs

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.
Cf. A363598, A363616, A363617, A363631.

Programs

  • Maple
    seq(coeff(series(add(x^k/(1+x^k)^4,k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    nmax = 44; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 44}]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*A000292(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)*(d + 2)/6.
a(n) = (4*A000593(n) + 6*A050999(n) + 2*A051000(n) - 2*A000203(n) - 3*A001157(n) - A001158(n))/6.
a(n) = (A138503(n) + 3*A321543(n) + 2*A002129(n)) / 6. - Amiram Eldar, Jan 04 2025

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).

A363631 Expansion of Sum_{k>0} (1/(1+x^k)^4 - 1).

Original entry on oeis.org

-4, 6, -24, 41, -60, 70, -124, 206, -244, 236, -368, 560, -564, 566, -896, 1175, -1144, 1180, -1544, 2042, -2168, 1942, -2604, 3650, -3336, 3100, -4304, 5096, -4964, 4940, -5988, 7720, -7528, 6636, -8616, 10809, -9884, 9126, -12064, 14548, -13248, 12796, -15184, 18192, -18412, 15830
Offset: 1

Views

Author

Seiichi Manyama, Jun 12 2023

Keywords

Links

Crossrefs

Cf. A363629, A363630.
Cf. A320901, A363598, A363616, A363617.

Programs

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

Formula

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

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