A008410 -id:A008410 - cp's OEIS frontend

A145154 Coefficients in expansion of Eisenstein series E_1.

1, 4, 8, 8, 12, 8, 16, 8, 16, 12, 16, 8, 24, 8, 16, 16, 20, 8, 24, 8, 24, 16, 16, 8, 32, 12, 16, 16, 24, 8, 32, 8, 24, 16, 16, 16, 36, 8, 16, 16, 32, 8, 32, 8, 24, 24, 16, 8, 40, 12, 24, 16, 24, 8, 32, 16, 32, 16, 16, 8, 48

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

			1 + 4*q + 8*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 16*q^6 + 8*q^7 + 16*q^8 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998

Cf. A000005, A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

with(numtheory); E:=proc(k) series(1-(2*k/bernoulli(k))*add( sigma[k-1](n)*q^n, n=1..60),q,61); end; E(1);

terms = 61; CoefficientList[1+4*Sum[x^k/(1-x^k), {k, 1, terms}]+O[x]^terms, x] (* Jean-François Alcover, Feb 27 2018 *)

{a(n) = if( n<1, n==0, 4 * numdiv(n))} /* Michael Somos, Jul 04 2011 */

a(0) = 1; for n >= 1, a(n) = 4*A000005(n). [After the PARI-program of Michael Somos.] - Antti Karttunen, May 25 2017

A282357 Coefficients in q-expansion of E_4^2*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1032, 48312, 171162336, -6444771144, -10105554483504, -1037089473751584, -48959817978105408, -1378102838778701640, -26186640301645703016, -364779940958775418032, -3952291567255306906464, -34798629548716507265568, -257403564989318828310384

Offset: 0

Seiichi Manyama, Feb 13 2017

sign

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

Cf. A008410 (E_4^2 = E_8), A058550 (E_4^2*E_6 = E_14), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^2*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]^2*E6[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282401 Eisenstein series E_28(q) (alternate convention E_14(q)), multiplied by 3392780147.

Original entry on oeis.org

3392780147, 6960, 934155393840, 53074158495516480, 125380214560150002480, 51856040954589843756960, 7123493021854278627673920, 457358042050198589771226240, 16828247534415852672059972400, 404722169541211889603611092720

Offset: 0

Seiichi Manyama, Feb 14 2017

nonn

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), A282356 (657931*E_26), this sequence (3392780147*E_28).

Cf. A282402 (E_4^7), A282403 (E_4^4*E_6^2), A282404 (E_4*E_6^4).

terms = 10;
E28[x_] = 3392780147 + 6960*Sum[k^27*x^k/(1 - x^k), {k, 1, terms}];
E28[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 489693897*A282402(n) + 2507636250*A282403(n) + 395450000*A282404(n).

A282402 Coefficients in q-expansion of E_4^7, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1680, 1224720, 505659840, 129351117840, 21060890131680, 2160822606183360, 134717272385473920, 4957295423282269200, 119288258695393463760, 2051465861242156554720, 26894077218337493424960, 281803532524538902825920

Offset: 0

Seiichi Manyama, Feb 14 2017

sign

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), A282330 (E_4^6), this sequence (E_4^7).

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^7 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A290009 Coefficients in expansion of 691E_4E_8*E_12.

Original entry on oeis.org

691, 563040, 305307360, 131729109120, 34085393629920, 4587384326495040, 302027782271806080, 10484303481804821760, 226150164335242994400, 3395290157453914541280, 38308806132696980919360, 343030311387007824977280, 2537869275676057371269760

Offset: 0

Seiichi Manyama, Jul 17 2017

nonn

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

Cf. A004009 (E_4), A008410 (E_8), A008411 (E_4^3), A029828 (691*E_12).

Cf. A290010.

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
691*E4[x]^3*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 691*E_4^3*E_12.

A294183 Coefficients in expansion of E_6/E_8.

Original entry on oeis.org

1, -984, 393768, -129252576, 38684099112, -10970838627984, 3003345011096352, -801909012374388672, 210169391033048138280, -54295810529811041175672, 13867098270790394508774768, -3508693915623201191415922848

Offset: 0

Seiichi Manyama, Feb 11 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A008410 (E_8). A013973 (E_6), A287933, A288840.

E_k/E_{k+2}: A294181 (k=2), A294182 (k=4), this sequence (k=6).

terms = 12;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E6[x]/E8[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Convolution inverse of A288840.

a(n) ~ (-1)^n * 512 * Pi^12 * exp(Pi*sqrt(3)*n) * n / (3 * Gamma(1/3)^18). - Vaclav Kotesovec, Jun 03 2018

A083728 a(0) = 1, a(n) = 480*sigma(n).

Original entry on oeis.org

1, 480, 1440, 1920, 3360, 2880, 5760, 3840, 7200, 6240, 8640, 5760, 13440, 6720, 11520, 11520, 14880, 8640, 18720, 9600, 20160, 15360, 17280, 11520, 28800, 14880, 20160, 19200, 26880, 14400, 34560, 15360, 30240, 23040, 25920, 23040, 43680, 18240, 28800

Offset: 0

N. J. A. Sloane, Jun 16 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A004009, A008410.

{1}~Join~Array[480 DivisorSigma[1, #] &, 38] (* Michael De Vlieger, Nov 16 2017 *)

A083728(n) = if(0==n,1,480*sigma(n)); \\ Antti Karttunen, Nov 16 2017

A282182 Eisenstein series E_30(q) (alternate convention E_15(q)), multiplied by 1723168255201.

Original entry on oeis.org

1723168255201, -171864, -92268782591832, -11795091175438423776, -49536425459206569762648, -32012164592742919922046864, -6332441368275869747902027488, -553385882817076320573218661312, -26594665913504249904864455466840

Offset: 0

Seiichi Manyama, Feb 16 2017

sign

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), A279893 (77683*E_22), A029831 (236364091*E_24), A282356 (657931*E_26), A282401 (3392780147*E_28), this sequence (1723168255201*E_30).

Cf. A282382 (E_4^6*E_6), A282461 (E_4^3*E_6^3), A282433 (E_6^5).

terms = 9;
E30[x_] = 1723168255201 - 171864*Sum[k^29*x^k/(1 - x^k), {k, 1, terms}];
E30[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 815806500201*A282382(n) + 881340705000*A282461(n) + 26021050000*A282433(n).

A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1920, 1630080, 803228160, 253366181760, 53205643249920, 7498254194403840, 699684356363412480, 42100628403784982400, 1614922125605880493440, 42332208491309728078080, 812648422343847344279040, 12060223533365891970132480

Offset: 0

Seiichi Manyama, Feb 16 2017

nonn

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), A282330 (E_4^6), A282402 (E_4^7), this sequence (E_4^8).

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^8 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282543 Coefficients in q-expansion of E_4^2*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1536, 551808, 163854336, -93387735168, -9709554816000, 4142226444876288, 642510156233453568, 41792421673548259200, 1615606968766288470528, 42343208407470359036160, 812663841518551604717568, 12060089370317565140003328

Offset: 0

Seiichi Manyama, Feb 17 2017

sign

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

Cf. A008410 (E_4^2 = E_8), A058550 (E_4^2*E_6 = E_14), A282292 (E_4^2*E_6^2 = E_10^2), A282357 (E_4^2*E_6^3), this sequence (E_4^2*E_6^4).

terms = 13;
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]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

cp's OEIS Frontend

A145154 Coefficients in expansion of Eisenstein series E_1.

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A282357 Coefficients in q-expansion of E_4^2*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282401 Eisenstein series E_28(q) (alternate convention E_14(q)), multiplied by 3392780147.

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A282402 Coefficients in q-expansion of E_4^7, where E_4 is the Eisenstein series A004009.

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A290009 Coefficients in expansion of 691*E_4*E_8*E_12.

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A294183 Coefficients in expansion of E_6/E_8.

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A083728 a(0) = 1, a(n) = 480*sigma(n).

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A282182 Eisenstein series E_30(q) (alternate convention E_15(q)), multiplied by 1723168255201.

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A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

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A290009 Coefficients in expansion of 691E_4E_8*E_12.