A363631 -id:A363631 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Oct 23 2018

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Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.

Cf. A363598, A363616, A363617, A363631.

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

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}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

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

Seiichi Manyama, Jun 12 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002129, A048272, A321543, A363629, A363631.

Cf. A320900, A363022, A363615.

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

a(n) = sumdiv(n, d, (-1)^d*binomial(d+2, 2));

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

A363629 Expansion of Sum_{k>0} (1/(1+x^k)^2 - 1).

Original entry on oeis.org

-2, 1, -6, 6, -8, 4, -10, 15, -16, 6, -14, 22, -16, 8, -28, 32, -20, 13, -22, 32, -36, 12, -26, 56, -34, 14, -44, 42, -32, 24, -34, 65, -52, 18, -52, 68, -40, 20, -60, 82, -44, 32, -46, 62, -84, 24, -50, 122, -60, 31, -76, 72, -56, 40, -76, 108, -84, 30, -62, 124, -64, 32, -110, 130, -88, 48, -70, 92

Offset: 1

Seiichi Manyama, Jun 12 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363630, A363631.

Cf. A002129, A048272, A325940.

a[n_] := DivisorSum[n, (-1)^#*(# + 1) &]; Array[a, 100] (* Amiram Eldar, Jul 18 2023 *)

PARI
```
a(n) = sumdiv(n, d, (-1)^d*(d+1));
```

G.f.: Sum_{k>0} (k+1) * (-x)^k/(1 - x^k).

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

cp's OEIS Frontend

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

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

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A363629 Expansion of Sum_{k>0} (1/(1+x^k)^2 - 1).

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