A289024 -id:A289024 - cp's OEIS frontend

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

504, 143388, 51180024, 20556578700, 8806299845112, 3929750661380124, 1803727445909594616, 845145871847732769804, 402283166289266872824312, 193877350835487271784566812, 94381548697864188120110027256, 46328820782943001597184984563596

Offset: 1

Seiichi Manyama, Jun 18 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

Cf. A288968 (k=2), A110163 (k=4), this sequence (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A013973 (E_6), A110163, A288840 (E_8/E_6), A289637.

a(n) = A013975(n^2) for n>=1.
a(n) = 12 + (1/(2*n)) * Sum_{d|n} A008683(n/d) * A288840(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289637(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 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

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

24, 348, 6424, 129300, 2778648, 62114524, 1428337176, 33527349924, 799482197272, 19302454317660, 470740035601176, 11575875047000596, 286650683468840472, 7140515309818664028, 178783562850377621272, 4496350112540599930692

Offset: 1

Seiichi Manyama, Jun 20 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..701

Cf. this sequence (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A006352 (E_2), A008683, A288877 (E_4/E_2), A289635.

a(n) = 2 + (1/(12*n)) * Sum_{d|n} A008683(n/d) * A288877(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289635(d).
a(n) ~ 1 / (n * r^(2*n)), where r = A057823. - Vaclav Kotesovec, Mar 08 2018

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

24, 196908, 42987544, 21974456220, 8544538312728, 3980088408377644, 1793770730037338136, 847156322106368439324, 401870774532436947447832, 193962999708079363021283628, 94363580764388112933729226776, 46332621615483591171320408201116

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4^2*E_6 = E_4*E_10 = E_6*E_8 = E_14.

Seiichi Manyama, Table of n, a(n) for n = 1..367

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), this sequence (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289640.

a(n) = 2 * A110163(n) + A288851(n) = A110163(n) + A289024(n) = A288851(n) + A288471(n) = 28 + (1/n) * (Sum_{d|n} A008683(n/d) * (2/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289640(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480q + 61920q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

-480, 53520, -8192480, 1417877520, -261761532384, 50337746997520, -9956715872256480, 2010450258635669520, -412391756829925376480, 85648872592091236716816, -17967933476075186380800480, 3800832540589574135423637520

Offset: 1

Seiichi Manyama, Jun 21 2017

sign

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), this sequence (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008410 (E_8), A008683, A288261 (E_10/E_8), A289638.

a(n) = 16 + (2/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d).
a(n) = 2 * A110163(n) = 2 * A013953(n^2). - Seiichi Manyama, Jun 22 2017
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289638(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ 2 * (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289639 Coefficients in expansion of -q*E'_10/E_10 where E_10 is the Eisenstein Series (A013974).

Original entry on oeis.org

264, 340560, 141251616, 85062410400, 43377095394864, 23729517350865216, 12591243615814264896, 6769208775901467246912, 3618692733697667332476264, 1939201752717876551124987360, 1038098212042387655796115897440

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..366

-q*E'_k/E_k: A289635 (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), this sequence (k=10), A289640 (k=14).

Cf. A006352 (E_2), A013974 (E_10), A285836, A289024.

nmax = 20; Rest[CoefficientList[Series[264*x*Sum[k*DivisorSigma[9, k]*x^(k-1), {k, 1, nmax}]/(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A289024(d).
a(n) = A288261(n)/3 + A288840(n)/2 + 20*A000203(n).
a(n) = -Sum_{k=1..n-1} A013974(k)*a(n-k) - A013974(n)*n.
G.f.: 1/3 * E_6/E_4 + 1/2 * E_8/E_6 - 5/6 * E_2.
a(n) ~ exp(2*Pi*n). - Vaclav Kotesovec, Jul 09 2017

A289294 Coefficients in expansion of E_10^(1/2).

Original entry on oeis.org

1, -132, -76428, -12686784, -4629945804, -1581036186312, -643032851554368, -264454897726360704, -114830224962140965068, -50847479367845783084484, -23070238839261012248537688, -10629338992044523324726971456

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367

E_k^(1/2): A289291 (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), this sequence (k=10), A289295 (k=14).

Cf. A013974 (E_10), A289024.

nmax = 20; s = 10; CoefficientList[Series[Sqrt[1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A289024(n)/2).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3^(3/2) * Pi^(5/2) / (2^(9/2) * Gamma(3/4)^12) = -0.3503612261281732359954402284478780636268623476628... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A289568 Coefficients in expansion of 1/E_10^(1/2).

Original entry on oeis.org

1, 132, 93852, 35163744, 18119136156, 8462089683432, 4234179302847648, 2096050696254014016, 1057219212439789539228, 534730176137991079392036, 272470142855167873443179352, 139363825115618499934478625696

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), this sequence (k=10), A289569 (k=14).

Cf. A285836 (1/E_10), A289024, A289294 (E_10^(1/2)).

nmax = 20; CoefficientList[Series[(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(-A289024(n)/2).

a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 0.4542595790370690606664796229968194763901027924111318430568304678613... = 2^(7/2) * Gamma(3/4)^12 / (3^(3/2) * Pi^(7/2)). - Vaclav Kotesovec, Jul 09 2017, updated Mar 07 2018

cp's OEIS Frontend

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504*q - 16632*q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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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) ...

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A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24*q - 72*q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24*q - 196632*q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480*q + 61920*q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289639 Coefficients in expansion of -q*E'_10/E_10 where E_10 is the Eisenstein Series (A013974).

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A289294 Coefficients in expansion of E_10^(1/2).

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A289568 Coefficients in expansion of 1/E_10^(1/2).

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A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A288968 Exponents a(1), a(2), ... such that E_2, 1 - 24q - 72q^2 - ... (A006352) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A288471 Exponents a(1), a(2), ... such that E_8, 1 + 480q + 61920q^2 + ... (A008410) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .