A008410 -id:A008410 - 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

A279892 Eisenstein series E_18(q) (alternate convention E_9(q)), multiplied by 43867.

Original entry on oeis.org

43867, -28728, -3765465144, -3709938631392, -493547047383096, -21917724609403728, -486272786232443616, -6683009405824511424, -64690198594597187640, -479102079577959825624, -2872821917728374840144, -14520482234727711482016, -63736746640768788267744

Offset: 0

Seiichi Manyama, Dec 22 2016

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J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), this sequence (43867*E_18), A029830 (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24).

Cf. A282000 (E_4^3*E_6), A282253 (E_6^3).

terms = 13;
E18[x_] = 43867 - 28728*Sum[k^17*x^k/(1 - x^k), {k, 1, terms}];
E18[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 43867 - 28728 * Sum_{i>=1} sigma_17(i)q^i where sigma_17(n) is A013965.

a(n) = 38367*A282000(n) + 5500*A282253(n). - Seiichi Manyama, Feb 11 2017

A279893 Eisenstein series E_22(q) (alternate convention E_11(q)), multiplied by 77683.

Original entry on oeis.org

77683, -552, -1157628456, -5774114968608, -2427722831757864, -263214111328125552, -12109202528761173024, -308317316973972772416, -5091303792066668003880, -60399282006368937251976, -552000263214112485753456, -4084937969230504375869024, -25394838301602325644596256, -136379620048544616772836528, -646588586243917921590531648

Offset: 0

Seiichi Manyama, Dec 22 2016

sign

J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), A279892 (43867*E_18), A029830 (174611*E_20), this sequence (77683*E_22), A029831 (236364091*E_24).

Cf. A282047 (E_4^4*E_6), A282328 (E_4*E_6^3).

terms = 15;
E22[x_] = 77683 - 552*Sum[k^21*x^k/(1 - x^k), {k, 1, terms}];
E22[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 77683 - 552 * Sum_{i>=1} sigma_21(i)q^i where sigma_21(n) is A013969.

a(n) = 57183*A282047(n) + 20500*A282328(n). - Seiichi Manyama, Feb 12 2017

A287933 Coefficients in expansion of 1/E_8.

Original entry on oeis.org

1, -480, 168480, -52199040, 15119446560, -4198347132480, 1132514464199040, -299116847254053120, 77742157641008378400, -19951615350261029163360, 5068304275307482667436480, -1276700988345016720650917760

Offset: 0

Seiichi Manyama, Jun 17 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A008410 (E_8).

Cf. A288816 (k=2), A001943 (k=4), A000706 (k=6), this sequence (k=8), A285836 (k=10), A287964 (k=14).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n, where c = (262144 * Pi^24) / (81 * Gamma(1/3)^36) = 1.0839091249080051624370140889296742679583925822413671... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A096961 a(n) = Sum_{0

Original entry on oeis.org

1, 128, 2188, 16384, 78126, 280064, 823544, 2097152, 4785157, 10000128, 19487172, 35848192, 62748518, 105413632, 170939688, 268435456, 410338674, 612500096, 893871740, 1280016384, 1801914272, 2494358016, 3404825448

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A007331, A008410, A013666, A096960, A096962, A096963.

A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* Michael Somos, Nov 30 2014 */

a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))};

ModularForms( Gamma0(2), 8, prec=24).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - Michael Somos, Jan 09 2015
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A282208 Coefficients in q-expansion of E_2^2*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 192, -8928, 9984, 1420896, 11433600, 53760384, 187233792, 533725920, 1327018944, 2953851840, 6060858624, 11611915392, 21030301824, 36387585792, 60357358080, 97020376032, 150755202432, 229107724704, 338493223680, 492378465600, 698632525824, 980953593984

Offset: 0

Seiichi Manyama, Feb 09 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006352 (E_2), A004009 (E_4), A281374 (E_2^2), A282019 (E_2*E_4), A008410 (E_4^2 = E_8), A282018 (E_2^3), this sequence (E_2^2*E_4), A282101 (E_2*E_4^2), A008411 (E_4^3).

terms = 23;
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]^2*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 432, 39960, -1418560, 17312940, -71928864, -462815680, 7500885120, -38038437810, 29000909200, 729783353376, -4661016429888, 13691625085880, -16503845217120, -14982974507520, 45085348093056, 99234456545637, -157805792764560, -1644659689877680

Offset: 2

Seiichi Manyama, Jul 23 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: A290178 (k=4), A290048 (k=6), this sequence (k=8), A290181 (k=10), A290182 (k=14).

Cf. A000594, A008410 (E_8).

terms = 19;
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E8[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_8, E_12 ; E_8, E_10, E_14 ; E_10, E_12, E_16]. G.f. is -691^2*3617*b(q)/(1728^2*2^3*3*5^3*7^2*467).

A282330 Coefficients in q-expansion of E_4^6, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1440, 876960, 292072320, 57349833120, 6660135541440, 436536302762880, 15172132360815360, 327295477379498400, 4913576699608450080, 55439481453769056960, 496426192564963006080, 3672749219557161663360, 23148323907214334109120

Offset: 0

Seiichi Manyama, Feb 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Eisenstein Series.

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), this sequence (E_4^6).

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^6 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^6.

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

A282356 Eisenstein series E_26(q) (alternate convention E_13(q)), multiplied by 657931.

Original entry on oeis.org

657931, -24, -805306392, -20334926626656, -27021598569529368, -7152557373046875024, -682326933054044766048, -32185646871935157619392, -906694391732570450559000, -17229551704624797057112632, -240000007152557373852181392

Offset: 0

Seiichi Manyama, Feb 13 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Eisenstein series

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), A279892 (43867*E_18), A029830 (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24), this sequence (657931*E_26).

Cf. A282048 (E_4^5*E_6), A282357 (E_4^2*E_6^3).

terms = 11;
E26[x_] = 657931 - 24*Sum[k^25*x^k/(1 - x^k), {k, 1, terms}];
E26[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 392931*A282048(n) + 265000*A282357(n).

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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A279892 Eisenstein series E_18(q) (alternate convention E_9(q)), multiplied by 43867.

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A279893 Eisenstein series E_22(q) (alternate convention E_11(q)), multiplied by 77683.

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A287933 Coefficients in expansion of 1/E_8.

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A096961 a(n) = Sum_{0

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A282208 Coefficients in q-expansion of E_2^2*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A282330 Coefficients in q-expansion of E_4^6, where E_4 is the Eisenstein series A004009.

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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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