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.

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

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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

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

A366937 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+1,2) * floor(n/k).

Original entry on oeis.org

1, -1, 6, -6, 10, -7, 22, -26, 26, -16, 51, -54, 38, -41, 101, -83, 71, -72, 119, -143, 123, -66, 211, -230, 111, -151, 279, -216, 220, -182, 315, -397, 237, -207, 467, -430, 274, -279, 599, -519, 343, -423, 524, -665, 557, -250, 879, -874, 380, -612, 874, -776
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Partial sums of A320900.
Cf. A024919, A366938, A366939.
Cf. A364970, A366395.

Programs

  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+1, 2)*(n\k));
  • Python
    from math import isqrt
def A366937(n): return (((s:=isqrt(m:=n>>1))*(s+1)**2*((s<<2)+5)<<1)-(t:=isqrt(n))*(t+1)**2*(t+2)-sum((((q:=m//w)+1)*(q*((q<<2)+5)+6*w*((w<<1)+1))<<1) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)+3*w*(w+1)) for w in range(1,t+1)))//6 # Chai Wah Wu, Oct 29 2023

Formula

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^3 = -1/(1-x) * Sum_{k>=1} binomial(k+1,2) * (-x)^k/(1-x^k).

A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -1, 9, -12, 20, -12, 35, -60, 72, -30, 77, -132, 104, -56, 210, -256, 170, -117, 209, -320, 378, -132, 299, -672, 425, -182, 594, -588, 464, -360, 527, -1040, 858, -306, 910, -1224, 740, -380, 1170, -1640, 902, -672, 989, -1364, 1890, -552, 1175, -2928, 1470, -775, 1938, -1872, 1484, -1080, 2090
Offset: 1

Views

Author

Seiichi Manyama, Jul 19 2023

Keywords

Links

Crossrefs

Cf. A320900, A364351.
Cf. A002129, A048272, A309731.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + 1)*(# + 1) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1+x^k)^3))

Formula

a(n) = (n/2) * Sum_{d|n} (-1)^(d+1) * (d+1) = (n/2) * (A002129(n) + A048272(n)).

A364351 Expansion of Sum_{k>0} k^2 * x^k/(1 + x^k)^3.

Original entry on oeis.org

1, 1, 15, -6, 40, 12, 77, -60, 180, 30, 187, -120, 260, 56, 630, -376, 442, 117, 551, -340, 1218, 132, 805, -1104, 1325, 182, 1998, -672, 1276, 360, 1457, -2032, 2970, 306, 3290, -1710, 2072, 380, 4134, -3080, 2542, 672, 2795, -1672, 7830, 552, 3337, -6816, 4998, 775, 7038, -2340, 4240, 1080
Offset: 1

Views

Author

Seiichi Manyama, Jul 19 2023

Keywords

Links

Crossrefs

Cf. A320900, A364343.
Cf. A000593, A048272, A309732.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(n/#+1) * (#+n) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k^2*x^k/(1+x^k)^3))

Formula

a(n) = (n/2) * Sum_{d|n} (-1)^(n/d+1) * (d+n) = (n/2) * (A000593(n) + n * A048272(n)).
Showing 1-5 of 5 results.

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