A377977 -id:A377977 - cp's OEIS frontend

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.

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

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

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

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

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

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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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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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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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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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.

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