A211342 -id:A211342 - 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

A289291 Coefficients in expansion of E_2^(1/2).

Original entry on oeis.org

1, -12, -108, -1344, -22044, -409752, -8201088, -172293504, -3746915388, -83625518604, -1904468689368, -44079484775616, -1033852665619200, -24518163456010392, -586936016770722048, -14164129272396668544, -344209494372831399036

Offset: 0

Seiichi Manyama, Jul 02 2017

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

E_k^(1/2): this sequence (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A006352 (E_2), A288968.

G.f.: Product_{n>=1} (1-q^n)^(A288968(n)/2).

a(n) ~ c / (r^n * n^(3/2)), where r = A211342 = 0.03727681029645165815098078... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.297340792206337929158904153045493466135450465337136... - Vaclav Kotesovec, Jul 02 2017

A288877 Coefficients in expansion of E_4/E_2.

Original entry on oeis.org

1, 264, 8568, 231456, 6214872, 166719024, 4472485344, 119980322880, 3218631807384, 86344077536616, 2316294684846288, 62137684699355232, 1666926011246777184, 44717506621139113584, 1199606572169515887552, 32181041313068138778816

Offset: 0

Seiichi Manyama, Jun 18 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..699

E_{k+2}/E_k: this sequence (k=2), A288261 (k=4), A288840 (k=6).
Cf. A004009 (E_4), A006352 (E_2), A288816 (1/E_2).
Cf. A211342.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 16; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[4]/Ei[2] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) ~ 12 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jun 28 2017

A288816 Coefficients in expansion of 1/E_2.

Original entry on oeis.org

1, 24, 648, 17376, 466152, 12505104, 335466144, 8999325120, 241418862504, 6476381979576, 173737557697968, 4660740989265312, 125030574027131424, 3354111390776151504, 89978497733627940672, 2413792838444465745216, 64753202305891291798824

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..700

Cf. A006352 (E_2).

Cf. this sequence (k=2), A001943 (k=4), A000706 (k=6), A287933 (k=8), A285836 (k=10), A287964 (k=14).

G.f.: 1/(1 - 24*sum(k>=1, k*x^k/(1 - x^k))).

a(n) ~ c / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.900080074462078245744608120875628441926356101483729... - Vaclav Kotesovec, Jul 02 2017

A289565 Coefficients in expansion of 1/E_2^(1/2).

Original entry on oeis.org

1, 12, 252, 5664, 133356, 3224952, 79387488, 1978996416, 49797787788, 1262193008556, 32177428972632, 824182154521056, 21193138994244960, 546767126418119352, 14146104826919725632, 366887630982365262144, 9535791498480146879436

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..701

1/E_k^(1/2): this sequence (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).

Cf. A288816 (1/E_2), A288968, A289291 (E_2^(1/2)).

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, 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)^(-A288968(n)/2).

a(n) ~ c / (sqrt(n) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.535261044779387956394739769118415667289349331646288208543596374426... - Vaclav Kotesovec, Jul 09 2017

A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

Original entry on oeis.org

24, 720, 19296, 517920, 13893264, 372707136, 9998360256, 268219317312, 7195339794744, 193024557070560, 5178140391612960, 138910500937231488, 3726458885094926160, 99967214347459657344, 2681753442755678231616

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

			a(1) = - A006352(1)*1 = 24,
a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.

Seiichi Manyama, Table of n, a(n) for n = 1..700

-q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.

nmax = 20; Rest[CoefficientList[Series[24*x*Sum[k*DivisorSigma[1, k]*x^(k-1), {k, 1, nmax}]/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A288968(d).
a(n) = A288877(n)/12 + 2*A000203(n).
a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.
G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.
a(n) ~ 1 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jul 09 2017

A377973 Expansion of the 96th root of the series 2*E_2(x) - E_2(x)^2, where E_2 is the Eisenstein series of weight 2.

Original entry on oeis.org

1, 0, -6, -36, -1812, -20748, -773340, -12237456, -386587650, -7368446268, -211914644940, -4517757977820, -123221458979940, -2814502962357420, -74551748141034552, -1778129476480366320, -46377354051910716180, -1137191336376638407704, -29438532048777299115090, -735051729258136807204140

Offset: 0

Peter Bala, Nov 13 2024

Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_2(x) lies in P(4) (Heninger et al.). Hence E_2(x)^2 lies in P(8).
We claim that the series 2*E_2(x) - E_2(x)^2 belongs to P(96).
Proof.
E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
Hence,
2*E_2(x) - E_2(x)^2 = 1 - (24^2)*(Sum_{n >= 1} sigma_1(n)*x^n )^2 is in the set R.
Hence, 2*E_2(x) - E_2(x)^2 == 1 (mod 24^2) == 1 (mod (2^6)*(3^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_2(x)^2 belongs to P((2^5)*3) = P(96). End Proof.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A006352 (E_2), A281374 (E_2)^2, A289392 ((E_2)^(1/4)), A341801, A341871 - A341875, A377974, A377975, A377976, A377977.

with(numtheory):
E := proc (k) local n, t1; t1 := 1 - 2*k*add(sigma[k-1](n)*q^n, n = 1..30)/bernoulli(k); series(t1, q, 30) end:
seq(coeftayl((2*E(2) - E(2)^2)^(1/96), q = 0, n),n = 0..20);

terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E2[x]^2)^(1/96), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c / (r^n * n^(97/96)), where r = A211342 = 0.03727681029645165815098... and c = -0.0104397599261506010365791466642760245638473040812140699981294533624... - Vaclav Kotesovec, Aug 03 2025

A057823 Decimal expansion of q = 0.193072033..., which is the value of q which gives the maximum of the Dedekind eta function eta(q) := q^(1/12) * Product_{n>=1} (1 - q^(2n)) for q between 0 and 1.

Original entry on oeis.org

1, 9, 3, 0, 7, 2, 0, 3, 3, 9, 5, 7, 4, 1, 0, 9, 7, 8, 9, 2, 2, 9, 4, 1, 6, 8, 5, 4, 2, 1, 2, 6, 2, 2, 5, 4, 5, 7, 0, 5, 0, 7, 7, 6, 0, 9, 7, 8, 7, 0, 4, 7, 2, 1, 6, 0, 9, 8, 0, 8, 9, 8, 9, 0, 7, 7, 7, 4, 6, 8, 4, 0, 5, 6, 7, 8, 7, 4, 9, 2, 5, 7, 0, 2, 8, 9, 6, 3, 9, 2, 7, 9, 3, 3, 6, 0, 8, 8, 0, 2

Offset: 0

Peter L. Walker (peterw(AT)aus.ac.ae), Nov 24 2000

			0.19307203395741097892294168542126225457050776097870...

Eric Weisstein's World of Mathematics, Dedekind Eta Function.

Cf. A211342.

RealDigits[FindRoot[D[q^(1/12)*Product[(1-q^(2 n)), {n, 100}], q] == 0, {q, 0.2}, WorkingPrecision -> 200][[1,2]]][[1]]
q /. Last @ FindMaximum[ DedekindEta[ -I*Log[q]/Pi], {q, 1/5}, WorkingPrecision -> 200] // RealDigits[#][[1]][[1 ;; 100]]&  (* Jean-François Alcover, Feb 05 2013 *)
q0 = q /. FindMaximum[q^(1/12)*QPochhammer[q^2], {q, 1/5}, WorkingPrecision -> 200][[2]]; RealDigits[q0, 10, 100][[1]] (* Jean-François Alcover, Nov 25 2015 *)

Equals sqrt(A211342). - Vaclav Kotesovec, Jul 02 2017

More terms from Vladeta Jovovic, Jun 19 2004

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

A289344 Coefficients in expansion of E_2^(1/2)/Product_{k>=1} (1-q^k).

Original entry on oeis.org

1, -11, -118, -1473, -23635, -434861, -8659573, -181387821, -3936961298, -87743843970, -1996149058302, -46163368994680, -1082012001849499, -25646334881233711, -613664275728573585, -14803437882920457712, -359626550280367615329

Offset: 0

Seiichi Manyama, Jul 03 2017

sign

Cf. A006352 (E_2), A288995, A289062, A289291.

Cf. A106205, A106203.

nmax = 20; CoefficientList[Series[Sqrt[1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]] / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 03 2017 *)

G.f.: Product_{k>=1} (1-q^k)^(A288995(k)/24).

a(n) ~ c / (n^(3/2) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.309300289625571303778321676728514880378401177270067457514896529... - Vaclav Kotesovec, Jul 03 2017

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

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A289291 Coefficients in expansion of E_2^(1/2).

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A288877 Coefficients in expansion of E_4/E_2.

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A288816 Coefficients in expansion of 1/E_2.

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A289565 Coefficients in expansion of 1/E_2^(1/2).

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A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

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A377973 Expansion of the 96th root of the series 2*E_2(x) - E_2(x)^2, where E_2 is the Eisenstein series of weight 2.

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A057823 Decimal expansion of q = 0.193072033..., which is the value of q which gives the maximum of the Dedekind eta function eta(q) := q^(1/12) * Product_{n>=1} (1 - q^(2n)) for q between 0 and 1.

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

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