A320901 -id:A320901 - 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).

A366938 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+2,3) * floor(n/k).

Original entry on oeis.org

1, -2, 9, -14, 22, -27, 58, -85, 91, -97, 190, -243, 213, -266, 460, -499, 471, -553, 778, -970, 896, -845, 1456, -1697, 1264, -1560, 2270, -2289, 2207, -2307, 3150, -3793, 3049, -3125, 4765, -5079, 4061, -4492, 6634, -6714, 5628, -6370, 7821, -9120, 7986, -7013

Offset: 1

Seiichi Manyama, Oct 29 2023

sign

Partial sums of A320901.
Cf. A024919, A366937, A366939.
Cf. A365409, A366659.

a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+2, 3)*(n\k));

from math import isqrt
def A366938(n): return (((s:=isqrt(m:=n>>1))*(s+1)**3*(s+2)<<4)-(t:=isqrt(n))*(t+1)**2*(t+2)*(t+3)-sum((((q:=m//w)+1)*(q*(q+1)*(q+2)+(w*(w+1)*((w<<1)+1)<<1))<<4) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)*(q+3)+(w*(w+1)*(w+2)<<2)) for w in range(1,t+1)))//24 # Chai Wah Wu, Oct 29 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^4 = -1/(1-x) * Sum_{k>=1} binomial(k+2,3) * (-x)^k/(1-x^k).

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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A366938 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+2,3) * floor(n/k).

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