A288968 -id:A288968 - cp's OEIS frontend

A288995 a(n) = 12 * (A288968(n) - 2).

264, 4152, 77064, 1551576, 33343752, 745374264, 17140046088, 402328199064, 9593786367240, 231629451811896, 5648880427214088, 138910500564007128, 3439808201626085640, 85686183717823968312, 2145402754204531455240, 53956201350487199168280

Offset: 1

Seiichi Manyama, Jun 23 2017

nonn

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

Related to E_{k+2}/E_k: this sequence (k=2), A192731 (k=4), A289061 (k=6).
Cf. A008683, A288877 (E_4*E_2), A288968.
Cf. A289062.

a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A288877(d).

A289394 a(n) = A288968(n)/4.

Original entry on oeis.org

6, 87, 1606, 32325, 694662, 15528631, 357084294, 8381837481, 199870549318, 4825613579415, 117685008900294, 2893968761750149, 71662670867210118, 1785128827454666007, 44695890712594405318, 1124087528135149982673, 28381310631267855206406

Offset: 1

Seiichi Manyama, Jul 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..702

Cf. A006352 (E_2), A288968, A289392 (E_2^(1/4)).

Cf. A289395, A289396.

a(n) = 1/2 + (1/(48*n)) * Sum_{d|n} A008683(n/d) * A288877(d).

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) ... .

Original entry on oeis.org

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

A289291 Coefficients in expansion of E_2^(1/2).

Original entry on oeis.org

1, -12, -108, -1344, -22044, -409752, -8201088, -172293504, -3746915388, -83625518604, -1904468689368, -44079484775616, -1033852665619200, -24518163456010392, -586936016770722048, -14164129272396668544, -344209494372831399036

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..702

E_k^(1/2): this sequence (k=2), A289292 (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A006352 (E_2), A288968.

G.f.: Product_{n>=1} (1-q^n)^(A288968(n)/2).

a(n) ~ c / (r^n * n^(3/2)), where r = A211342 = 0.03727681029645165815098078... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = -0.297340792206337929158904153045493466135450465337136... - Vaclav Kotesovec, Jul 02 2017

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

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

264, 170148, 47083784, 21265517460, 8675419078920, 3954919534878884, 1798749087973466376, 846151096977050604564, 402076970410851910136072, 193920175271783317402925220, 94372564731126150526919627016, 46330721199213296384252696382356

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4*E_6 = E_10.

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), this sequence (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289639.

a(n) = A110163(n) + A288851(n) = 20 + (1/n) * (Sum_{d|n} A008683(n/d) * (1/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289639(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

A289565 Coefficients in expansion of 1/E_2^(1/2).

Original entry on oeis.org

1, 12, 252, 5664, 133356, 3224952, 79387488, 1978996416, 49797787788, 1262193008556, 32177428972632, 824182154521056, 21193138994244960, 546767126418119352, 14146104826919725632, 366887630982365262144, 9535791498480146879436

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..701

1/E_k^(1/2): this sequence (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).

Cf. A288816 (1/E_2), A288968, A289291 (E_2^(1/2)).

nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, 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)^(-A288968(n)/2).

a(n) ~ c / (sqrt(n) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.535261044779387956394739769118415667289349331646288208543596374426... - Vaclav Kotesovec, Jul 09 2017

A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

Original entry on oeis.org

24, 720, 19296, 517920, 13893264, 372707136, 9998360256, 268219317312, 7195339794744, 193024557070560, 5178140391612960, 138910500937231488, 3726458885094926160, 99967214347459657344, 2681753442755678231616

Offset: 1

Seiichi Manyama, Jul 09 2017

nonn

			a(1) = - A006352(1)*1 = 24,
a(2) = -(A006352(1)*a(1)) - A006352(2)*2 = 720,
a(3) = -(A006352(1)*a(2)  + A006352(2)*a(1)) - A006352(3)*3 = 19296,
a(4) = -(A006352(1)*a(3)  + A006352(2)*a(2)  + A006352(3)*a(1)) - A006352(4)*4 = 517920.

Seiichi Manyama, Table of n, a(n) for n = 1..700

-q*E'_k/E_k: this sequence (k=2), A289636 (k=4), A289637 (k=6), A289638 (k=8), A289639 (k=10), A289640 (k=14).

Cf. A000203, A006352 (E_2), A076835, A211342, A288816, A288877, A288968.

nmax = 20; Rest[CoefficientList[Series[24*x*Sum[k*DivisorSigma[1, k]*x^(k-1), {k, 1, nmax}]/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)

a(n) = Sum_{d|n} d * A288968(d).
a(n) = A288877(n)/12 + 2*A000203(n).
a(n) = -Sum_{k=1..n-1} A006352(k)*a(n-k) - A006352(n)*n.
G.f.: 1/12 * E_4/E_2 - 1/12 * E_2.
a(n) ~ 1 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jul 09 2017

cp's OEIS Frontend

A288995 a(n) = 12 * (A288968(n) - 2).

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A289394 a(n) = A288968(n)/4.

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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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A289291 Coefficients in expansion of E_2^(1/2).

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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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A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264*q - 135432*q^2 + ... (A013974) 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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A289565 Coefficients in expansion of 1/E_2^(1/2).

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Mathematica

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A289635 Coefficients in expansion of -q*E'_2/E_2 where E_2 is the Eisenstein Series (A006352).

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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) ... .

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) ... .

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) 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) ... .