A110163 -id:A110163 - cp's OEIS frontend

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

24, 348, 6424, 129300, 2778648, 62114524, 1428337176, 33527349924, 799482197272, 19302454317660, 470740035601176, 11575875047000596, 286650683468840472, 7140515309818664028, 178783562850377621272, 4496350112540599930692

Offset: 1

Seiichi Manyama, Jun 20 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..701

Cf. this sequence (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A006352 (E_2), A008683, A288877 (E_4/E_2), A289635.

a(n) = 2 + (1/(12*n)) * Sum_{d|n} A008683(n/d) * A288877(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289635(d).
a(n) ~ 1 / (n * r^(2*n)), where r = A057823. - Vaclav Kotesovec, Mar 08 2018

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

24, 196908, 42987544, 21974456220, 8544538312728, 3980088408377644, 1793770730037338136, 847156322106368439324, 401870774532436947447832, 193962999708079363021283628, 94363580764388112933729226776, 46332621615483591171320408201116

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4^2*E_6 = E_4*E_10 = E_6*E_8 = E_14.

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), this sequence (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289640.

a(n) = 2 * A110163(n) + A288851(n) = A110163(n) + A289024(n) = A288851(n) + A288471(n) = 28 + (1/n) * (Sum_{d|n} A008683(n/d) * (2/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289640(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

264, 170148, 47083784, 21265517460, 8675419078920, 3954919534878884, 1798749087973466376, 846151096977050604564, 402076970410851910136072, 193920175271783317402925220, 94372564731126150526919627016, 46330721199213296384252696382356

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4*E_6 = E_10.

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), this sequence (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289639.

a(n) = A110163(n) + A288851(n) = 20 + (1/n) * (Sum_{d|n} A008683(n/d) * (1/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289639(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289308 Coefficients in expansion of E_4^(3/8).

Original entry on oeis.org

1, 90, -5940, 758520, -115431930, 19355028840, -3447208777320, 639751846440960, -122326632902618100, 23925871041887048130, -4763590542726586318440, 962102309316632909723880, -196619722885250960565506040, 40580696990507644723354537320

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..424

E_4^(k/8): A108091 (k=1), A289307 (k=2), this sequence (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(3/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 3^(7/4) * Gamma(1/3)^(27/4) / (64 * 2^(3/8) * Pi^(9/2) * Gamma(5/8)) = 0.2574920621515873836544977885672468081360882154861344422709504189964... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A289309 Coefficients in expansion of E_4^(5/8).

Original entry on oeis.org

1, 150, -5400, 625200, -86672550, 13570016400, -2289741037200, 406440122001600, -74830416797043000, 14162747887897808550, -2738995393669565720400, 538973037306449327998800, -107578899914865970323788400, 21729813219122500082762389200

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), this sequence (k=5), A289318 (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(5/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 5 * 3^(5/4) * Gamma(1/3)^(45/4) / (256 * 2^(5/8) * Pi^(15/2) * Gamma(3/8)) = 0.2571085249207580781634342667473393997795373224370302803101380883544... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289319 Coefficients in expansion of E_4^(7/8).

Original entry on oeis.org

1, 210, -1260, 232680, -28907970, 4211355960, -671557897080, 113817372354240, -20151698294479500, 3687092782592216970, -692109989731133096760, 132609267059636375116920, -25838624519733523814390760, 5105657091664960508653858680

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

In general, for 0 < m < 1, the expansion of (E_4)^m is asymptotic to (-1)^(n+1) * m * 3^(2*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*n) / (2^(9*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), this sequence (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(7/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(7*A110163(n)/8).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 7 * 3^(7/4) * Gamma(1/3)^(63/4) / (1024 * 2^(7/8) * Pi^(21/2) * Gamma(1/8)) = 0.1121182787986009012644546699220584282491804117887058146553161217384... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480q + 61920q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

-480, 53520, -8192480, 1417877520, -261761532384, 50337746997520, -9956715872256480, 2010450258635669520, -412391756829925376480, 85648872592091236716816, -17967933476075186380800480, 3800832540589574135423637520

Offset: 1

Seiichi Manyama, Jun 21 2017

sign

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), this sequence (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008410 (E_8), A008683, A288261 (E_10/E_8), A289638.

a(n) = 16 + (2/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d).
a(n) = 2 * A110163(n) = 2 * A013953(n^2). - Seiichi Manyama, Jun 22 2017
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289638(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ 2 * (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289318 Coefficients in expansion of E_4^(3/4).

Original entry on oeis.org

1, 180, -3780, 447840, -59046660, 8921092680, -1463828444640, 253953515257920, -45858209756343300, 8534765953624978260, -1626301691950399586280, 315807346469727624396960, -62284193156782292089690080, 12443904711281870749228431240

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), this sequence (k=6), A289319 (k=7).

Cf. A004009 (E_4), A110163.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,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*A110163(n)/4).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 3^(5/2) * Gamma(1/3)^(27/2) / (256 * 2^(3/4) * Pi^9 * Gamma(1/4)) = 0.2007048471908800363193160136812560289856774734680572658944418664975... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289636 Coefficients in expansion of -q*E'_4/E_4 where E_4 is the Eisenstein Series (A004009).

Original entry on oeis.org

-240, 53280, -12288960, 2835808320, -654403831200, 151013228757120, -34848505552897920, 8041801037378486400, -1855762905734676483120, 428244362959801779806400, -98823634118413525094402880, 22804995243537595828606337280

Offset: 1

Seiichi Manyama, Jul 09 2017

sign

			a(1) = 1 * A110163(1) = -240,
a(2) = 1 * A110163(1) + 2 * A110163(2) = 53280,
a(3) = 1 * A110163(1) + 3 * A110163(3) = -12288960.

Seiichi Manyama, Table of n, a(n) for n = 1..422

-q*E'_k/E_k: A289635 (k=2), this sequence (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A001943, A004009 (E_4), A006352 (E_2), A110163, A145094, A289633.

nmax = 20; Rest[CoefficientList[Series[-240*x*Sum[k*DivisorSigma[3, k]*x^(k-1), {k, 1, nmax}]/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)
terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[-D[Ei[4], x]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = Sum_{d|n} d * A110163(d) = A289633(n)/6.
a(n) = A288261(n)/3 + 8*A000203(n).
a(n) = -Sum_{k=1..n-1} A004009(k)*a(n-k) - A004009(n)*n.
G.f.: 1/3 * E_6/E_4 - 1/3 * E_2.
a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jul 09 2017

A289566 Coefficients in expansion of 1/E_4^(1/2).

Original entry on oeis.org

1, -120, 20520, -3934560, 793510440, -164694615120, 34824089129760, -7460017581785280, 1613575314347164200, -351613291994820018840, 77073167391611232305520, -16975579813113940564868640, 3753822590560913900129106720

Offset: 0

Seiichi Manyama, Jul 08 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..423

1/E_k^(1/2): A289565 (k=2), this sequence (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).

Cf. A001943 (1/E_4), A110163, A289292 (E_4^(1/2)).

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/2).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / sqrt(n), where c = 3^(7/2) * Gamma(2/3)^9 / (2^(9/2) * Pi^(7/2)) = 0.5756695813762774104492155417156662666189119445257965... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

cp's OEIS Frontend

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24*q - 72*q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24*q - 196632*q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264*q - 135432*q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289308 Coefficients in expansion of E_4^(3/8).

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A289309 Coefficients in expansion of E_4^(5/8).

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A289319 Coefficients in expansion of E_4^(7/8).

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A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480*q + 61920*q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289318 Coefficients in expansion of E_4^(3/4).

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A289636 Coefficients in expansion of -q*E'_4/E_4 where E_4 is the Eisenstein Series (A004009).

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A289566 Coefficients in expansion of 1/E_4^(1/2).

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A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480q + 61920q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .