A363617 -id:A363617 - 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

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).

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

Seiichi Manyama, Jun 11 2023

sign

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

Cf. A363022, A363617.

Cf. A363608.

a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 4] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^5)))

a(n) = sumdiv(n, d, (-1)^d*binomial(d, 4));

G.f.: Sum_{k>0} binomial(k,4) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d,4).

cp's OEIS Frontend

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

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

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

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