A363614 -id:A363614 - cp's OEIS frontend

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378

Offset: 1

Seiichi Manyama, Jun 11 2023

sign

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

Cf. A325940, A363598, A363613, A363614.

Cf. A002129, A069153, A321543.

a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))

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

G.f.: Sum_{k>0} binomial(k,2) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.

A363598 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.

Original entry on oeis.org

0, 1, -4, 11, -20, 32, -56, 95, -124, 146, -220, 328, -364, 400, -584, 775, -816, 881, -1140, 1486, -1600, 1552, -2024, 2712, -2620, 2562, -3400, 4064, -4060, 4112, -4960, 6231, -6208, 5730, -7216, 8947, -8436, 8000, -10248, 12230, -11480, 11232, -13244, 15752

Offset: 1

Seiichi Manyama, Jun 11 2023

sign

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

Cf. A325940, A363022, A363613, A363614.

Cf. A002129, A138503, A363604.

a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^4)))

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

G.f.: Sum_{k>0} binomial(k+1,3) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d+1,3) = (A002129(n) - A138503(n))/6.

A363613 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^5.

Original entry on oeis.org

0, 1, -5, 16, -35, 66, -126, 226, -335, 461, -715, 1082, -1365, 1695, -2420, 3286, -3876, 4581, -5985, 7791, -8986, 9912, -12650, 16242, -17585, 19111, -24086, 29115, -31465, 34106, -40920, 49662, -53080, 55030, -66206, 79412, -82251, 85406, -102640, 119931

Offset: 1

Seiichi Manyama, Jun 11 2023

sign

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

Cf. A325940, A363022, A363598, A363614.

Cf. A363605.

a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^5)))

a(n) = sumdiv(n, d, (-1)^d*binomial(d+2, 4));

G.f.: Sum_{k>0} binomial(k+2,4) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d+2,4).

cp's OEIS Frontend

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

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A363598 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.

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A363613 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^5.

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