A108091 -id:A108091 - cp's OEIS frontend

A341871 Coefficients of the series whose 48th power equals E_2(x)^2/E_4(x), where E_2(x) and E_4(x) are the Eisenstein series A006352 and A004009.

1, -6, 558, -88884, 15433662, -2864048616, 552921962724, -109731286565040, 22220439670517814, -4569456313225317114, 951159953810624453208, -199945837161334089352548, 42373766861587365894611604

Offset: 0

Peter Bala, Feb 22 2021

It is easy to see that E_2(x)^2/E_4(x) == 1 - 48*Sum_{k >= 1} (k + 5*k^3)*x^k/(1 - x^k) (mod 288), and also that the integer k + 5*k^3 is always divisible by 6. Hence, E_2(x)^2/E_4(x) == 1 (mod 288). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_4(x))^(1/48) = 1 - 6*x + 558*x^2 - 88884*x^3 + 15433662*x^4 - ... has integer coefficients.

Note that (E_2(x)^2/E_4(x))^(1/48) = (E_2(x)^4/E_8(x))^(1/96).

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.
Wikipedia, Eisenstein series

Cf. A004009, A006352, A108091, A289392, A341872, A341873, A341874, A341875.

E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)^2/E(4,x))^(1/48), x, 20):
seriestolist(%);

A341875 Coefficients of the series whose 24th power equals E_2(x)*E_4(x)/E_6(x), where E_2(x), E_4(x) and E_6(x) are the Eisenstein series A006352, A004009 and A013973.

Original entry on oeis.org

1, 30, 5310, 2453220, 910100190, 409796742600, 181276113779460, 84362079365838960, 39636500385830239350, 18986938020443181757410, 9186944625290601368703000, 4491611148118819794144792660, 2212757749022582852433835771860, 1097546094982154634980848454416920

Offset: 0

Peter Bala, Feb 23 2021

Since E_2(x)*E_4(x)/E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 - 21*k^5)*x^k/(1 - x^k) (mod 144), and since the integer k - 10*k^3 - 21*k^5 is always divisible by 6 it follows that E_2(x)*E_4(x)/E_6(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x)/E_6(x))^(1/24) = 1 + 30*x + 5310*x^2 + 2453220*x^3 + 910100190*x^4 + ... has integer coefficients.
From Peter Bala, Nov 16 2024 (Start):
Expansion of ( E_2(x)*E_8(x)/E_10(x) )^(1/24), where E_k(x) is the Eisenstein series of weight k.
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) and E_10(x) lie in P(4) while the series E_8(x) lies in P(16) (Heninger et al.).
 We claim that the series (E_2(x)*E_8(x))/E_10(x) belongs to P(24).
Proof.
E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
E_8(x) = 1 + 480*Sum_{n >= 1} sigma_7(n)*x^n.
E_10(x) = 1  - 264*Sum_{n >= 1} sigma_9(n)*x^n.
Hence, E_2(x)*E_8(x)/E_10(x) == 1 + (12^2)*Sum_{n >= 1} (1/6)*(-sigma_1(n) + 20*sigma_7(n) + 11*sigma_9(n))*x^n (mod 12^2) in R. The polynomial (1/6)*(-k + 20*k^7 + 11*k^9) of degree 9 is integer-valued since it takes integer values for 10 consective values of n (e.g., from n = 0 to n = 9).
Hence, E_2(x)*E_8(x)/E_10(x) == 1 (mod 12^2) == 1 (mod (2^4)*(3^2)) in R.
It follows from Heninger et al., Theorem 1, Corollary 2, that the series E_2(x)*E_8(x)/E_10(x) belongs to P((2^3)*3) = P(24). End Proof. (End)

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.
Wikipedia, Eisenstein series

Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8), A013973, A013974 (E_10). A108091 (E_8)^(1/16), A110150 ((E_10)^(1/4)), A289392 ((E_2)^(1/4)), A341871 - A341874, A377973, A377974, A377975, A377976, A377977.

E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)*E(4,x)/E(6,x))^(1/24), x, 20):
seriestolist(%);

a(n) ~ c * exp(2*Pi*n) / n^(23/24), where c = 0.0431061156115657949750305669836959595841497962033916083447436... - Vaclav Kotesovec, Mar 08 2021

Equals the series ( E_2(x)*E_8(x)/E_10(x) )^(1/24). - Peter Bala, Nov 16 2024

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

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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

A289247 Coefficients in expansion of 1/E_4^(1/8).

Original entry on oeis.org

1, -30, 3780, -616440, 111056910, -21135698280, 4165203862440, -840914061328320, 172810940671692900, -35998781800053352710, 7579904611028433074280, -1609957152292592382408360, 344417407415742189796786680, -74127324674775434904036905640

Offset: 0

Seiichi Manyama, Jul 08 2017

sign

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

E_4^(k/8): A001943 (k=-8), A289566 (k=-4), A295815 (k=-2), this sequence (k=-1), A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8).

Cf. A110163, A300147.

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

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/8).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = Pi^(3/2) / (2^(15/8) * 3^(1/4) * Gamma(1/3)^(9/4) * Gamma(9/8)) = 0.133402757019143151407904538533... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018

A294974 Coefficients in expansion of (E_2^4/E_4)^(1/8).

Original entry on oeis.org

1, -42, 4032, -659904, 118064226, -22406634432, 4407587356032, -888750999070464, 182478248639753472, -37986867560948245674, 7994272624037726124672, -1697243410477799687716416, 362963150140702802158191360, -78095916585903527021840348352

Offset: 0

Seiichi Manyama, Feb 12 2018

sign

Also coefficients in expansion of (E_2^8/E_8)^(1/16).

Cf. A004009 (E_4), A006352 (E_2), A108091, A289247, A289291, A294626, A294976.

terms = 14;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
(E2[x]^4/E4[x])^(1/8) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: Product_{n>=1} (1-q^n)^A294626(n).

a(n) ~ (-1)^n * 2^(13/8) * Pi * exp(Pi*sqrt(3)*n) / (Gamma(1/8) * Gamma(1/3)^(9/4) * n^(7/8)). - Vaclav Kotesovec, Jun 03 2018

A377974 Expansion of the 1920th root of the series 2*E_4(x) - E_8(x), where E_4 and E_8 are the Eisenstein series of weight 4 and weight 8.

Original entry on oeis.org

1, 0, -30, -540, -867660, -31107300, -33668157900, -1795572812400, -1477793386682970, -103845834995498100, -69550699526934273180, -6017200267937951322660, -3426636160378174348594500, -349303370036461528632524580, -174458882971934188146144343320, -20314204536496741742949242177040

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_4(x) lies in P(8) (Heninger et al.). Since E_8(x) = E_4(x)^2, it follows that E_8(x) lies in P(16).
We claim that the series 2*E_4(x) - E_8(x) belongs to P(1920).
Proof.
E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n. Hence,
2*E_4(x) - E_8(x) = 2*E_4(x) - E_4(x)^2 = 1 - 240^2*( Sum_{n >= 1} sigma_3(n) )^2 is in the set R.
Hence, 2*E_4(x) - E_8(x) == 1 mod(240^2) == 1 (mod (2^8)*(3^2)*(5^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_4(x) - E_8(x) belongs to P((2^7)*3*5) = P(1920). 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), A008410 (E_8), A108091 (eighth root of E_4), A341871 - A341875, A377973, 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(4) - E(8))^(1/1920), q = 0, n),n = 0..20);

terms = 20; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E4[x] - E8[x])^(1/1920), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c / (r^n * n^(1921/1920)), where r = 0.004019427095115250686492968205049012182922598389629390919504184161606551652... is the root of the equation Sum_{k>=1} sigma_3(k) * r^k = 1/240 and c = -0.00052087420429807426289253718287... - 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

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

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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).

A303007 Expansion of (1-240*x)^(1/8).

Original entry on oeis.org

1, -30, -3150, -472500, -81506250, -15160162500, -2956231687500, -595469525625000, -122815589660156250, -25791273828632812500, -5493541325498789062500, -1183608449221102734375000, -257434837705589844726562500, -56437637496994696728515625000

Offset: 0

Seiichi Manyama, Jun 15 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A003557, A007947, A049210, A108091.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), this sequence (b=240), A303055 (b=504), A305886 (b=1728).

CoefficientList[Series[Surd[1-240x,8],{x,0,20}],x] (* Harvey P. Dale, Aug 29 2024 *)

N=20; x='x+O('x^N); Vec((1-240*x)^(1/8))

a(n) = 30^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = 15^n * A301271(n).
a(n) ~ -2^(4*n - 3) * 15^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +30*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A377976 Expansion of the 48th root of the series 2*E_2(x) - E_4(x), where E_2(x) and E_4(x) are the Eisenstein series of weight 2 and 4.

Original entry on oeis.org

1, -6, -894, -174420, -38431614, -9048710040, -2221653118116, -561444889080960, -144914324838755910, -38011797621225586602, -10098281618881696696392, -2710458654395655881518356, -733711171629600485187568404, -200033609249999737396399900920, -54867682197669353983111639906656

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_2(x) lies in P(4) and E_4(x) lies in P(8) (Heninger et al.).
We claim that the series 2*E_2(x) - E_4(x) belongs to P(48).
Proof.
E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n.
Hence,
2*E_2(x) - E_4(x) = 1 - (288)*Sum_{n >= 1} ((1/6)*sigma_1(n) + (5/6)*sigma_3(n))*x^n belongs to the set R, since the polynomial (1/6)*k + (5/6)*k^3 has integer values for integer k. See A004068.
Hence, 2*E_2(x) - E_4(x) == 1 (mod 288) == 1 (mod (2^5)*(3^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_4(x) belongs to P((2^4)*3) = P(48). End Proof.
In a similar way we find that the series 3*E_2(x) - E_6(x) - 1 belongs to P(72) and the three series 3*E_4(x) - 2*E_6(x), 5*E_4(x) - 2*E_10(x) - 2 and 5*E_6(x) - 3*E_10(x) - 1 belong to P(288).

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. A004068, A006352 (E_2), A004009 (E_4), A108091 ((E_4)^1/8), A289392 ((E_2)^(1/4)), A341871 - A341875, A377973, A377974, A377975, 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(4))^(1/48), q = 0, n),n = 0..20);

terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E4[x])^(1/48), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c * d^n / n^(49/48), where d = 295.8669385406700495308385233671383399895922733900742171390678012914822364544611... and c = -0.0205882497833853345146399243734199945444083043388859856935627869352251231763... - Vaclav Kotesovec, Aug 03 2025

cp's OEIS Frontend

A341871 Coefficients of the series whose 48th power equals E_2(x)^2/E_4(x), where E_2(x) and E_4(x) are the Eisenstein series A006352 and A004009.

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A341875 Coefficients of the series whose 24th power equals E_2(x)*E_4(x)/E_6(x), where E_2(x), E_4(x) and E_6(x) are the Eisenstein series A006352, A004009 and A013973.

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

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A289247 Coefficients in expansion of 1/E_4^(1/8).

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A294974 Coefficients in expansion of (E_2^4/E_4)^(1/8).

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A377974 Expansion of the 1920th root of the series 2*E_4(x) - E_8(x), where E_4 and E_8 are the Eisenstein series of weight 4 and weight 8.

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

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A303007 Expansion of (1-240*x)^(1/8).

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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.