A110150 -id:A110150 - 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

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

A289391 Coefficients in expansion of E_14^(1/4).

Original entry on oeis.org

1, -6, -49212, -10451544, -4218246978, -1581565900392, -677142351901080, -293172823731286848, -132241381826055031692, -60651805300034501958126, -28350123351848675673466968, -13420046900399367136336144200

Offset: 0

Seiichi Manyama, Jul 05 2017

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

E_k^(1/4): A289392 (k=2), A289307 (k=4), A289326 (k=6), A289292 (k=8), A110150 (k=10), this sequence (k=14).

Cf. A004984, A058550 (E_14).

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[13, 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)^(A289029(n)/4).
a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3*Pi^2 / (2^(17/4) * Gamma(3/4)^9) = -0.2497407198517688195944362279691013167903920989625478927175764401875... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018
G.f.: Sum_{k>=0} A004984(k) * (3*f(q))^k where f(q) is Sum_{k>=1} sigma_13(k)*q^k. - Seiichi Manyama, Jun 16 2018

A295788 Coefficients in expansion of (E_10/E_2^10)^(1/4).

Original entry on oeis.org

1, -6, -41652, -11504904, -4378103178, -1652544433080, -700184843900712, -302796005909941632, -136251754253507319300, -62421509259448987324542, -29147951871527035454309160, -13787807362002100397282325912

Offset: 0

Seiichi Manyama, Feb 13 2018

sign

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

Cf. A110150, A294976, A294978, A299712.

terms = 12;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
(E10[x]/E2[x]^10)^(1/4) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ -Pi^4 * exp(2*Pi*n) / (3^(7/4) * 2^(15/4) * Gamma(3/4)^7 * n^(5/4)). - Vaclav Kotesovec, Jun 03 2018

A295790 Coefficients in expansion of E_10^(-1/4).

Original entry on oeis.org

1, 66, 44748, 14628504, 7092895062, 3108317470632, 1487551265488728, 706997530642862976, 344758210436694126204, 169166701402985932594410, 83918648754791525856013272, 41891672444283862758775998792, 21045241902624298023560126068200

Offset: 0

Seiichi Manyama, Feb 13 2018

nonn

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

Cf. A013974 (E_10), A110150.

Convolution inverse of A110150.

a(n) ~ 2^(5/4) * 3^(1/4) * Gamma(3/4)^7 * exp(2*Pi*n) / (3 * Pi^(5/2) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

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

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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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A289391 Coefficients in expansion of E_14^(1/4).

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A295788 Coefficients in expansion of (E_10/E_2^10)^(1/4).

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A295790 Coefficients in expansion of E_10^(-1/4).

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