A013974 -id:A013974 - 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

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

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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 *)

A282382 Coefficients in q-expansion of E_4^6*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 936, 134568, -173988576, -104617833048, -27210540914064, -3910401774129888, -322823174243838912, -15429983442476298840, -469709326015243815672, -9973673112569954220432, -158215072218253260221088, -1972939697011615168926432

Offset: 0

Seiichi Manyama, Feb 16 2017

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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), A282048 (E_4^5*E_6), this sequence (E_4^6*E_6).

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]^6*E6[x] + 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).

A282461 Coefficients in q-expansion of E_4^3*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -792, -197208, 180534816, 34731625896, -11282282306064, -3475192229286624, -319729598062193088, -15436589476561121880, -469831003553540798136, -9973761497118317484432, -158213220814147434639264, -1972935965978751882433248

Offset: 0

Seiichi Manyama, Feb 16 2017

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

Cf. A013974 (E_4*E_6 = E_10), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^3*E_6^3 = E_10^3).

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

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

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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).

A282404 Coefficients in q-expansion of E_4*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1776, 975888, -66529344, -79516693488, 9511628122080, 2031621786790848, 134911299030780288, 4962883791154433040, 119289719378991436368, 2051366007318600561120, 26893975935849646148928, 281804567385216854182848

Offset: 0

Seiichi Manyama, Feb 14 2017

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

Cf. A013974 (E_4*E_6 = E_10), A282287 (E_4*E_6^2), A282328 (E_4*E_6^3), this sequence (E_4*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]*E6[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A290049 Coefficients in expansion of 691E_8E_10*E_12.

Original entry on oeis.org

691, 214776, 10042488, -31595258016, -37453557900168, -14820419119618224, -2593285239712936608, -222297419357081232192, -10663770067272328258440, -324599563661107722245352, -6891830216922929182318512, -109326152051786546417315808

Offset: 0

Seiichi Manyama, Jul 19 2017

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

Cf. A008410 (E_8), A013974 (E_10), A029828 (691*E_12).

terms = 12;
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E10[x_] = 1 - 264*Sum[k^9*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*E8[x]*E10[x]*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 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

A282540 Eisenstein series E_32(q) (alternate convention E_16(q)), multiplied by 7709321041217.

Original entry on oeis.org

7709321041217, 32640, 70093866303360, 20160859654708062720, 150525431711563807489920, 151991844177246093750032640, 43295116458269350559666465280, 5149788469617367127914995164160, 323250903208723929093223124860800

Offset: 0

Seiichi Manyama, Feb 17 2017

nonn

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), A282182 (1723168255201*E_30), this sequence (7709321041217*E_32).

Cf. A282474 (E_4^8), A282541 (E_4^5*E_6^2), A282543 (E_4^2*E_6^4).

terms = 9;
E32[x_] = 7709321041217 + 32640*Sum[k^31*x^k/(1 - x^k), {k, 1, terms}];
E32[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

a(n) = 764412173217*A282474(n) + 5323905468000 * A282541(n) + 1621003400000 * A282543(n).

cp's OEIS Frontend

A145154 Coefficients in expansion of Eisenstein series E_1.

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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.

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A282382 Coefficients in q-expansion of E_4^6*E_6, 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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A282461 Coefficients in q-expansion of E_4^3*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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

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A282404 Coefficients in q-expansion of E_4*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A290049 Coefficients in expansion of 691*E_8*E_10*E_12.

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

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A282540 Eisenstein series E_32(q) (alternate convention E_16(q)), multiplied by 7709321041217.

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A290049 Coefficients in expansion of 691E_8E_10*E_12.