A363630 -id:A363630 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Oct 23 2018

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

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

Cf. A363022, A363615, A363630.

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

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}]

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

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

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

Seiichi Manyama, Jun 12 2023

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

Cf. A363629, A363630.

Cf. A320901, A363598, A363616, A363617.

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

a(n) = sumdiv(n, d, (-1)^d*binomial(d+3, 3));

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).

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

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

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

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

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