A289394 -id:A289394 - cp's OEIS frontend

A289392 Coefficients in expansion of E_2^(1/4).

1, -6, -72, -1104, -20238, -405792, -8601840, -189317568, -4281478272, -98841343686, -2318973049008, -55118876238000, -1324194430710912, -32099173821105312, -784045854628721568, -19276683937074656064, -476644852188898489662

Offset: 0

Seiichi Manyama, Jul 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..702

E_2^(k/4): this sequence (k=1), A289291 (k=2), A289393 (k=3).
E_k^(1/4): this sequence (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), A289391 (k=14).
Cf. A004984, A006352 (E_2), A108091, A109817, A289394.

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^A289394(n).
a(n) ~ c / (n^(5/4) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.209452682241344640265132676904094736935029272937832600102950644347... - Vaclav Kotesovec, Jul 08 2017
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_1(k)*q^k. - Seiichi Manyama, Jun 16 2018

A289395 a(n) = A110163(n)/8.

Original entry on oeis.org

-30, 3345, -512030, 88617345, -16360095774, 3146109187345, -622294742016030, 125653141164729345, -25774484801870336030, 5353054537005702294801, -1122995842254699148800030, 237552033786848383463977345, -50601782105721473281984512030

Offset: 1

Seiichi Manyama, Jul 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 1..424

Cf. A004009 (E_4), A108091 (E_4^(1/8)), A110163.

Cf. A289394, A289396.

a(n) = 1 + (1/(24*n)) * Sum_{d|n} A008683(n/d) * A288261(d).

A289396 a(n) = A288851(n)/12.

Original entry on oeis.org

42, 11949, 4265002, 1713048225, 733858320426, 327479221781677, 150310620492466218, 70428822653977730817, 33523597190772239402026, 16156445902957272648713901, 7865129058155349010009168938, 3860735065245250133098748713633

Offset: 1

Seiichi Manyama, Jul 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367

Cf. A013973 (E_6), A109817 (E_6^(1/12)), A288851.

Cf. A289394, A289395.

a(n) = 1 + (1/(24*n)) * Sum_{d|n} A008683(n/d) * A288840(d).

A289393 Coefficients in expansion of E_2^(3/4).

Original entry on oeis.org

1, -18, -108, -936, -13194, -224424, -4218264, -84318336, -1759467636, -37903487130, -836893437912, -18844318997496, -431163494289720, -9997357777073064, -234430475682110256, -5550426839122171776, -132513976699508759994

Offset: 0

Seiichi Manyama, Jul 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..703

E_2^(k/4): A289392 (k=1), A289291 (k=2), this sequence (k=3).

Cf. A006352 (E_2), A289394.

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(3/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A289394(n)).

a(n) ~ c / (n^(7/4) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.22385630328806525639758543854251232523806175231599584032442913209... - Vaclav Kotesovec, Jul 08 2017

cp's OEIS Frontend

A289392 Coefficients in expansion of E_2^(1/4).

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A289395 a(n) = A110163(n)/8.

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A289396 a(n) = A288851(n)/12.

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A289393 Coefficients in expansion of E_2^(3/4).

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