A013973 -id:A013973 - cp's OEIS frontend

A282048 Coefficients in q-expansion of E_4^5*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

1, 696, -34632, -167186976, -64422848328, -11387712944304, -1037073232984608, -48892286706157632, -1378097272692189000, -26188038166214133672, -364779879415169299632, -3952277018332870144608, -34798618196377082329632, -257403706082325167732976

Offset: 0

Seiichi Manyama, Feb 05 2017

sign

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.

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

Cf. A004009 (E_4), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), A282000 (E_4^3*E_6), A282047 (E_4^4*E_6), this sequence (E_4^5*E_6).

Cf. A037947, A281959, A281979.

terms = 14;
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]^5*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

-24 * A281959(n) = 657931 * a(n) - 457920000 * A037947(n) for n > 0.

A282332 Coefficients in q-expansion of E_4^3*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -288, -325728, 11700864, 35176468896, 6601058210880, 438061091013504, 15173572442740992, 327251435243536800, 4913611331706352224, 55439979246339307200, 496425441863436557184, 3672747479405396310912, 23148319784349233726784

Offset: 0

Seiichi Manyama, Feb 12 2017

sign

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

Cf. A280869 (E_6^2), A282287 (E_4*E_6^2), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^3*E_6^2).

terms = 14;
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]^3*E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282595 Coefficients in q-expansion of E_2^2*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -552, 7992, 460896, -3450504, -88161264, -728085024, -3775195968, -14894175240, -48567693576, -137214605232, -347495426784, -804758753568, -1733365307184, -3511286411328, -6753825302976, -12422812497672, -21971174382288, -37567247938344

Offset: 0

Seiichi Manyama, Feb 19 2017

sign

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

Cf. A282018 (E_2^3), this sequence (E_2^2*E_6), A282576 (E_2*E_6^2), A282253 (E_6^3).

terms = 19;
E2[x_] = 1 - 24*Sum[k*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]^2*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282596 Coefficients in q-expansion of E_2E_4^2E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -48, -196128, -33542976, -678319104, 12136422240, 509314518144, 7469015889792, 68272650653760, 458377820557584, 2454769903187520, 11035857376010304, 43103740076823552, 149954656815201504, 473331019057949952, 1375248429330791040, 3719662610125117632

Offset: 0

Seiichi Manyama, Feb 19 2017

sign

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

Cf. A282102 (E_2*E_4*E_6), A282547 (E_2*E_4*E_6^2).

terms = 17;
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]^2 *E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282547 Coefficients in q-expansion of E_2E_4E_6^2, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -792, -648, 67840416, 3219716376, 16790031216, -1536150710304, -39898324202688, -522122582192040, -4650999065751096, -31648313780323632, -175516685804469024, -827282698744164768, -3413275186936731984, -12598131165680789568, -42296014044574387776

Offset: 0

Seiichi Manyama, Feb 18 2017

sign

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

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

terms = 16;
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]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282780 Coefficients in q-expansion of E_2^3*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Original entry on oeis.org

1, -576, 21168, 308736, -15034608, -39208320, 1590712128, 20299281408, 137107250640, 665776675008, 2599125524640, 8637331788288, 25350641846208, 67336913702016, 164742803455104, 376186503674880, 809848148403024, 1657081821679488, 3243133560510576

Offset: 0

Seiichi Manyama, Feb 21 2017

sign

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

Cf. A282096 (E_2*E_6), A282595 (E_2^2*E_6), this sequence (E_2^3*E_6).

terms = 19;
E2[x_] = 1 - 24*Sum[k*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]^3*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282792 Coefficients in q-expansion of E_2^2E_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, -312, -122328, 1193376, 120735336, 123318576, -26119268064, -383848045248, -3132125965080, -18290795499096, -84925855577232, -331983655889184, -1133781877844448, -3470165144567184, -9697162366507968, -25093220330304576, -60786860467926552

Offset: 0

Seiichi Manyama, Feb 21 2017

sign

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

Cf. A282102 (E_2*E_4*E_6), this sequence (E_2^2*E_4*E_6), A282596 (E_2*E_4^2*E_6), A282547 (E_2*E_4*E_6^2).

terms = 17;
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*E6[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A281371 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 0, 1, 36, 492, 3608, 18828, 74760, 250352, 717984, 1866558, 4365580, 9635472, 19639032, 38559416, 71222616, 128258496, 219619968, 370366101, 597550068, 955638824, 1471571136, 2253173892, 3335433368, 4932972864, 7064391840, 10133162774, 14128072488, 19743952032, 26864847352

Offset: 0

N. J. A. Sloane, Feb 04 2017

nonn

This is (up to a constant factor), the numerator of the expression phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016; also Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

Cf. A006352, A004009, A013973, A145094, A281372 (the square root).

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)^2/518400,q,M+1);
seriestolist(t1);

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}];
(E2[x]*E4[x] - E6[x])^2/518400 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A281373 Coefficients in q-expansion of (E_2E_4 - E_6)^2/(300(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488

Offset: 0

N. J. A. Sloane, Feb 04 2017

nonn

This is (up to a constant factor), the function phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016.
Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

Cf. A006352, A004009, A013973, A145094, A281371 (the numerator), A000594 (the denominator), A319134, A319294.

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)^2/518400,q,M+1);
t2:=series((e4^3-e6^2)/1728,q,M+1);
t3:=series(t1/t2,q,M+1);
seriestolist(t3);

terms = 22;
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])^2/(300*(E6[x]^2 - E4[x]^3)) + O[x]^terms // CoefficientList[#, x]& // Abs (* Jean-François Alcover, Feb 27 2018 *)

a(n) ~ exp(4*Pi*sqrt(n)) / (14400 * sqrt(2) * Pi^2 * n^(7/4)). - Vaclav Kotesovec, Jun 06 2018

A282328 Coefficients in q-expansion of E_4*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1272, 351432, 89559456, -28689603384, -3415837464144, -155926897275744, -3967939206760128, -65540990858009400, -777517458842153496, -7105797244669716432, -52584588767807410464, -326903749149928526688, -1755591468945924647184

Offset: 0

Seiichi Manyama, Feb 12 2017

sign

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

Cf. A004009 (E_4), A013973 (E_6).

Cf. A013974 (E_4*E_6 = E_10), A282287 (E_4*E_6^2), this sequence (E_4*E_6^3).

terms = 14;
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]*E6[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

cp's OEIS Frontend

A282048 Coefficients in q-expansion of E_4^5*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A282332 Coefficients in q-expansion of E_4^3*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282595 Coefficients in q-expansion of E_2^2*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282596 Coefficients in q-expansion of E_2*E_4^2*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282547 Coefficients in q-expansion of E_2*E_4*E_6^2, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282780 Coefficients in q-expansion of E_2^3*E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282792 Coefficients in q-expansion of E_2^2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A281371 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A281373 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/(300*(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A282596 Coefficients in q-expansion of E_2E_4^2E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282547 Coefficients in q-expansion of E_2E_4E_6^2, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282792 Coefficients in q-expansion of E_2^2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A281373 Coefficients in q-expansion of (E_2E_4 - E_6)^2/(300(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.