A109817 -id:A109817 - cp's OEIS frontend

A289348 Coefficients in expansion of E_6^(5/6).

1, -420, -31500, -4724160, -1314429900, -440028142344, -162555920654400, -63990327056960640, -26341675849615282380, -11210298679649742846180, -4895195936831699458605912, -2181913188022929464292248640

Offset: 0

Seiichi Manyama, Jul 03 2017

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E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), this sequence (k=10), A289349 (k=11).

Cf. A013973 (E_6), A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(5/6), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A288851(n)/6).

a(n) ~ c * exp(2*Pi*n) / n^(11/6), where c = -5 * 3^(1/6) * Gamma(1/4)^(40/3) / (2048*sqrt(2) * Pi^(19/2) * Gamma(1/3)^2) = -0.1571123439957640423587958439875289712533650298096956968521099309872... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289349 Coefficients in expansion of E_6^(11/12).

Original entry on oeis.org

1, -462, -24948, -2518824, -654112074, -212483064024, -76819071738024, -29728723632736128, -12066341379893331300, -5073593348593538950566, -2192302482140061697816872, -968086916154014421082349304, -435126775136273350146250044888

Offset: 0

Seiichi Manyama, Jul 03 2017

sign

In general, for 0 < m < 1, the expansion of (E_6)^m is asymptotic to -m * Gamma(1/4)^(16*m) * 3^(2*m) * exp(2*Pi*n) / (2^(13*m) * Pi^(12*m) * Gamma(1-m) * n^(1+m)). - Vaclav Kotesovec, Mar 05 2018

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), this sequence (k=11).

Cf. A013973 (E_6), A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(11/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(11*A288851(n)/12).

a(n) ~ c * exp(2*Pi*n) / n^(23/12), where c = -11 * 2^(5/12) * 3^(5/6) * Pi^(11/3) / (128 * Gamma(1/12) * Gamma(3/4)^(44/3)) = -0.08406022472181281739983743854923746657261382508944840919197295490535... - Vaclav Kotesovec, Jul 08 2017

A289567 Coefficients in expansion of 1/E_6^(1/2).

Original entry on oeis.org

1, 252, 103572, 46355904, 21754545876, 10493652271032, 5153897870227008, 2563741466120209536, 1287429765611338091988, 651251466581383330576956, 331360676706818772917367912, 169399388595923901462013678656

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), this sequence (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
E_6^(k/12): A289570 (k=-18), A000706 (k=-12), this sequence (k=-6), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A000706 (1/E_6), A288851, A289293 (E_6^(1/2)).

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,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)^(-A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 2^(5/2) * Gamma(3/4)^8 / (3*Pi^(5/2)) = 0.5480868931611627439175185425300450785609564636925943866686455998197... - Vaclav Kotesovec, Jul 09 2017, updated Mar 03 2018

A110150 G.f.: 4th root of Eisenstein series E_10 (cf. A013974).

Original entry on oeis.org

1, -66, -40392, -9009264, -3725341158, -1400292801072, -604993149612720, -262280205541007808, -118717180239835505592, -54520207050101542651506, -25525844887805197307977968, -12095360676632550886664063760, -5797006133905562955666277287792, -2803076705590018145443840156918512

Offset: 0

N. J. A. Sloane, Sep 15 2005

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Vaclav Kotesovec, Table of n, a(n) for n = 0..360
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

E_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), this sequence (k=10), A289391 (k=14).

Cf. A004984, A013974, A109817, A289294.

nmax = 20; s = 10; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3^(3/4) * Pi^(3/2) / (2^(15/4) * Gamma(3/4)^7) = -0.227361380713650977567497769428903183591275821407342369621... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

G.f.: Sum_{k>=0} A004984(k) * (33*f(q))^k where f(q) is Sum_{k>=1} sigma_9(k)*q^k. - Seiichi Manyama, Jun 16 2018

A294976 Coefficients in expansion of (E_6/E_2^6)^(1/12).

Original entry on oeis.org

1, -30, -11340, -3912600, -1520905170, -636170644008, -278687199310200, -126000360658968000, -58290111778749466140, -27440829122946510954630, -13096614404248661886145848, -6320198941502349713305002120, -3077986352751848627729986859400

Offset: 0

Seiichi Manyama, Feb 12 2018

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Cf. A109817, A289565, A294974, A294975.

terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E6[x]/E2[x]^6)^(1/12) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: Product_{n>=1} (1-q^n)^A294975(n).
a(n) ~ -Gamma(1/3)^2 * Gamma(1/4)^(10/3) * exp(2*Pi*n) / (16 * 2^(1/12) * 3^(7/12) * Pi^(5/2) * Gamma(1/12) * n^(13/12)). - Vaclav Kotesovec, Jun 03 2018
Equivalently, a(n) ~ -Gamma(1/3) * Gamma(1/4)^(7/3) * exp(2*Pi*n) / (2^(23/6) * 3^(23/24) * Pi^2 * sqrt(1 + sqrt(3)) * n^(13/12)). - Vaclav Kotesovec, Nov 26 2024

A303055 Expansion of (1-504*x)^(1/12).

Original entry on oeis.org

1, -42, -9702, -3124044, -1148086170, -453264419916, -187198205425308, -79746435511181208, -34749509273997211386, -15405615778138763714460, -6923283730695560413278324, -3145688371456037358687733032, -1442298118312593128958325595172

Offset: 0

Seiichi Manyama, Jun 15 2018

Seiichi Manyama, Table of n, a(n) for n = 0..371

Cf. A109817.

N=20; x='x+O('x^N); Vec((1-504*x)^(1/12))

a(n) = 42^n/n! * Product_{k=0..n-1} (12*k - 1) for n > 0.
a(n) ~ -504^n / (12 * Gamma(11/12) * n^(13/12)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +42*(-12*n+13)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A377975 Expansion of the 6048th root of the series 2*E_6(x) - E_6(x)^2, where E_6 is the Eisenstein series of weight 6.

Original entry on oeis.org

1, 0, -42, -2772, -5399688, -704781084, -943173698460, -180121119486672, -188146584694894350, -46293152603021155692, -40574254265781269371884, -11963000065787771567311500, -9221266403646163252100062068, -3107813621461888912485774582588, -2176998806586925223600540321844120

Offset: 0

Peter Bala, Nov 14 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_6(x) lies in P(12) (Heninger et al.).
We claim that the series 2*E_6(x) - E_6(x)^2 belongs to P(6048).
Proof.
E_6(x) = 1 - 504*Sum_{n >= 1} sigma_5(n)*x^n. Hence,
2*E_6(x) - E_6(x)^2 = 1 - (504^2)*( Sum_{n >= 1} sigma_5(n)*x^n )^2 is in R.
Hence, 2*E_6(x) - E_6(x)^2 == 1 (mod 504^2) == 1 (mod (2^6)*(3^4)*(7^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_6(x) - E_6(x)^2 belongs to P((2^5)*(3^3)*7) = P(6048). End Proof.

Cf. A013973 (E_6), A109817 ( (E_6)^1/12 ), A280869 (E_6)^2, A341871 - A341875, A377973, A377974, 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(6) - E(6)^2)^(1/6048), q = 0, n),n = 0..20);

terms = 20; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E6[x] - E6[x]^2)^(1/6048), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c / (r^n * n^(6049/6048)), where r = 0.0018674427317079888144302129348270303934228050024753171993815386383179351229... is the root of the equation Sum_{k>=1} sigma_5(k) * r^k = 1/504 and c = -0.0001653486643613776568861731992670297686378824546... - Vaclav Kotesovec, Aug 03 2025

A377977 Expansion of the 288th root of the series 3E_4(x) - 2E_6(x), where E_4(x) and E_6(x) are the Eisenstein series of weight 4 and 6.

Original entry on oeis.org

1, 6, -5028, 5704188, -7284893010, 9926715853068, -14092613175928308, 20580782244716567592, -30684764269418402550900, 46478269075227117026711730, -71284154421570122590465786956, 110437754516732491586466670733772, -172528135408494997625486967978486588, 271418933884659782820559630827037837908

Offset: 0

Peter Bala, Nov 15 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_4(x) lies in P(8) and E_6(x) lies in P(12) (Heninger et al.).
We claim that the series 3*E_4(x) - 2*E_6(x) belongs to P(288).
Proof.
E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n.
E_6(x) = 1 - 504*Sum_{n >= 1} sigma_5(n)*x^n.
Hence, 3*E_4(x) - 2*E_6(x) = 1 - (12^3)*Sum_{n >= 1} (1/12)*(5*sigma_3(n) + 7*sigma_5(n))*x^n belong to R, since the polynomial (1/12)*(5*k^3 + 7*k^5) is integral for integer values of k. See A245380.
Hence, 3*E_4(x) - 2*E_6(x) == 1 (mod 12^3) == 1 (mod (2^6)*(3^3)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 3*E_4(x) - 2*E_6(x) belongs to P((2^5)*(3^2)) = P(288). 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. A004009 (E_4), A013973 (E_6), A108091 ((E_4)^1/8), A109817 ((E_6)^1/12), A245380, A341871 - A341875, A377973, A377974, A377975.

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((3*E(4) - 2*E(6))^(1/288), q = 0, n), n = 0..20);

terms = 20; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(3*E4[x] - 2*E6[x])^(1/288), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ (-1)^(n+1) * c * d^n / n^(289/288), where d = 1704.7780406875645261102091212390097973945883014209828800432529862899259963... and c = 0.0034650713031853295969588514070741337333119867976967661391075146399616... - Vaclav Kotesovec, Aug 03 2025

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).

A289570 Coefficients in expansion of 1/E_6^(3/2).

Original entry on oeis.org

1, 756, 501228, 311671584, 187266950892, 110121960638088, 63808586297102304, 36578013578688141504, 20797655630223547290348, 11749541312124028845092052, 6603568491137827506152966712, 3695593478842608407829235523808

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

E_6^(k/12): this sequence (k=-18), A000706 (k=-12), A289567 (k=-6), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-3/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-3*A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) * sqrt(n), where c = 2^(17/2) * Gamma(3/4)^24 / (27 * Pi^(13/2)) = 1.0344943380746471723299237298670710161068814236907171661035... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

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A289348 Coefficients in expansion of E_6^(5/6).

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A289349 Coefficients in expansion of E_6^(11/12).

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A289567 Coefficients in expansion of 1/E_6^(1/2).

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A110150 G.f.: 4th root of Eisenstein series E_10 (cf. A013974).

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A294976 Coefficients in expansion of (E_6/E_2^6)^(1/12).

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A303055 Expansion of (1-504*x)^(1/12).

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A377975 Expansion of the 6048th root of the series 2*E_6(x) - E_6(x)^2, where E_6 is the Eisenstein series of weight 6.

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A377977 Expansion of the 288th root of the series 3*E_4(x) - 2*E_6(x), where E_4(x) and E_6(x) are the Eisenstein series of weight 4 and 6.

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

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A377977 Expansion of the 288th root of the series 3E_4(x) - 2E_6(x), where E_4(x) and E_6(x) are the Eisenstein series of weight 4 and 6.