A004009 -id:A004009 - cp's OEIS frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

1, 30, -2880, 416640, -69178110, 12378401280, -2321610157440, 449733567736320, -89200812128140800, 18013245273252679710, -3689479088922151082880, 764375901202388789804160, -159862757100127037505991680, 33699694000689939789618455040, -7152050326608893289997995966720, 1526705794390267864554876727856640

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 06 2005

sign

			More precisely, the theta series of E_8 begins 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + ... and the 8th root of this is 1 + 30*q^2 - 2880*q^4 + 416640*q^6 - 69178110*q^8 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..424
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.
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.
N. J. A. Sloane, Seven Staggering Sequences.

E_4^(k/8): this sequence (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5).

Cf. A001158, A004009 (E_4), A106205, A110163, A300147, A303007.

nmax = 20; s = 8; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/16), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

R. = PowerSeriesRing(ZZ,20)
a = R(eisenstein_series_qexp(4,20, normalization='integral'))
list(a.sqrt().sqrt().sqrt()) # Andy Huchala, Jul 10 2021

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/8). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 3^(1/4) * Gamma(1/3)^(9/4) / (2^(33/8) * Pi^(3/2) * Gamma(7/8)) = 0.1141392450598624077174159151600898926678394937157356242319309115... - Vaclav Kotesovec, Jul 02 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
G.f.: Sum_{k>=0} A303007(k) * (-f(q))^k where f(q) is Sum_{k>=1} sigma_3(k)*q^k. - Seiichi Manyama, Jun 15 2018

A110163 Exponents a(1), a(2), ... such that theta series of E_8 lattice, 1 + 240 q + 2160 q^2 + ... (A004009) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ...

Original entry on oeis.org

-240, 26760, -4096240, 708938760, -130880766192, 25168873498760, -4978357936128240, 1005225129317834760, -206195878414962688240, 42824436296045618358408, -8983966738037593190400240, 1900416270294787067711818760, -404814256845771786255876096240, 86744167089111545378556727322760

Offset: 1

N. J. A. Sloane, Sep 16 2005

sign

Negative of inverse Euler transform of [240, 2160, ...].

			From _Seiichi Manyama_, Jun 17 2017: (Start)
a(1) = 8 + 1/3 * A008683(1/1) * A288261(1) = 8 + 1/3 * (-744) = -240,
a(2) = 8 + 1/6 * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = 8 + 1/6 * (744 + 159768) = 26760. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..424

Cf. A288968 (k=2), this sequence (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A004009, A008683, A013953, A288261, A289636.

terms = 14; Clear[a, sol];
a4009[n_] := If[n == 0, 1, 240 DivisorSigma[3, n]];
sol[0] = {}; sol[kmax_] := sol[kmax] = Join[sol[kmax-1], SolveAlways[ Sum[ a4009[k] q^k, {k, 0, kmax}] == Normal[Product[(1-q^k)^a[k], {k, 1, kmax}] + O[q]^(kmax+1)] /. sol[kmax-1], q][[1]] ];
A110163 = Array[a, terms] /. sol[terms] (* Jean-François Alcover, Jul 03 2017 *)

a(n) = A013953(n^2) for n>=1. - Seiichi Manyama, Jun 17 2017
a(n) = 8 + (1/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d). - Seiichi Manyama, Jun 17 2017
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289636(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400

Offset: 0

N. J. A. Sloane, Feb 04 2017

The q-expansion of the square of this expression is given in A281371.

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

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

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

[0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

with(gfun):
with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)/720,q,M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024

Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

Original entry on oeis.org

1, 960, 354240, 61543680, 4858169280, 137745912960, 2120861041920, 21423820362240, 158753769048000, 928983317334720, 4512174992346240, 18847874280625920, 69518972236842240, 230951926208599680, 701949379778818560, 1975788826748167680

Offset: 0

Seiichi Manyama, Feb 04 2017

nonn

Also coefficients in q-expansion of E_8^2.

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 207.

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), this sequence (E_4^4), A282015 (E_4^5).
Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), this sequence (E_8^2), A282292 (E_10^2).
Cf. A013963, A027364, A281876.

terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^4.

16320 * A013963(n) = 3617 * a(n) - 3456000 * A027364(n) for n > 0.

A282019 Coefficients in q-expansion of E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

Original entry on oeis.org

1, 216, -3672, -62496, -322488, -1121904, -2969568, -6737472, -13678200, -24978312, -43826832, -70620768, -112325472, -166558896, -248342976, -346320576, -491604984, -655461072, -897864696, -1154109600, -1532856528, -1921344768, -2488726944, -3042415296, -3876616800, -4639932504

Offset: 0

N. J. A. Sloane, Feb 05 2017

sign

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

Cf. A004009, A006352.

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
series(e2*e4,q,M+1);
seriestolist(%);

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

A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 258, 6564, 66052, 390630, 1693512, 5764808, 16909320, 43066413, 100782540, 214358892, 433565328, 815730734, 1487320464, 2564095320, 4328785936, 6975757458, 11111134554, 16983563060, 25801892760, 37840199712, 55304594136, 78310985304, 110992776480

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), this sequence (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A013955, A013666, A196288.

terms = 25;
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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^8*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018

G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282101(n) - A013974(n))/720. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
G.f. Sum_{k>=1} k^8*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

			a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
Cf. A001160, A013664, A131472.

terms = 30;
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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

a(n) = A145095(n)/504 for n > 0.
G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -288, -129168, -1927296, 65152656, 1535768640, 15223408704, 98001292032, 474055120080, 1870878793824, 6312358836000, 18835985199744, 50831420617152, 126257508465984, 292348744636032, 637474437331200, 1319883180896592, 2610964045674432, 4963491913583664

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

The series expansion of the 12th root of the generating function gives A341801. - Peter Bala, Feb 23 2021

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

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A013974 (E_10).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), A282101 (E_2*E_8), this sequence (E_2*E_10), A341801.

terms = 19;
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}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

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

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
Cf. A222171.

[0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-3).
Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

Original entry on oeis.org

1, 456, 50328, -470496, -21784008, -234371664, -1446514848, -6502690752, -23328111240, -71276388312, -191952331632, -468159788448, -1052750026272, -2212261706256, -4394299104576, -8303419066176, -15060718806024, -26284654025712, -44471780630856

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

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

Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), this sequence (E_2*E_8), A282102 (E_2*E_10).

terms = 19;
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]*E4[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

cp's OEIS Frontend

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A110163 Exponents a(1), a(2), ... such that theta series of E_8 lattice, 1 + 240 q + 2160 q^2 + ... (A004009) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A282019 Coefficients in q-expansion of E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.