A288877 -id:A288877 - cp's OEIS frontend

A294626 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288877(d) - A288261(d)).

42, -3171, 515242, -88552695, 16361485098, -3146078130083, 622295456184618, -125653124401054383, 25774485201611434666, -5353054527354475135971, 1122995842490069166600618, -237552033781060445940477047, 50601782105864798623718932266

Offset: 1

Seiichi Manyama, Feb 12 2018

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

Cf. A008683, A288261, A288877, A294974.

terms = 13;
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}];
a[n_] := (1/(24 n))*Sum[MoebiusMu[n/d]*SeriesCoefficient[E4[x]/E2[x] - E6[x]/E4[x], {x, 0, d}], {d, Divisors[n]}];
Array[a, terms] (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ -(-1)^n * exp(Pi*sqrt(3)*n) / (8*n). - Vaclav Kotesovec, Jun 03 2018

A294975 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

Original entry on oeis.org

30, 11775, 4261790, 1712983575, 733856931102, 327479190724415, 150310619778297630, 70428822637214055855, 33523597190372498303390, 16156445902947621421555071, 7865129058155113639991368350, 3860735065245244345161225213335

Offset: 1

Seiichi Manyama, Feb 12 2018

nonn

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

Cf. A008683, A288840, A288877, A294976.

terms = 12;
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}];
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
a[n_] := (1/(24 n))*Sum[MoebiusMu[n/d]*SeriesCoefficient[E8[x]/E6[x] - E4[x]/E2[x], {x, 0, d}], {d, Divisors[n]}];
Array[a, terms] (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ exp(2*Pi*n) / (12*n). - Vaclav Kotesovec, Jun 03 2018

A288261 Coefficients in expansion of E_6/E_4.

Original entry on oeis.org

1, -744, 159768, -36866976, 8507424792, -1963211493744, 453039686271072, -104545516658693952, 24125403112135458840, -5567288717204029449672, 1284733088879405339418768, -296470902355240575283208928, 68414985730612787485819011168

Offset: 0

Seiichi Manyama, Jun 17 2017

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Also coefficients in expansion of E_10/E_8. - Seiichi Manyama, Jun 20 2017

			G.f.: 1 - 744*q + 159768*q^2 - 36866976*q^3 + 8507424792*q^4 - 1963211493744*q^5 + 453039686271072*q^6 + ...
From _Seiichi Manyama_, Jun 27 2017: (Start)
a(0) = j_0((-1+sqrt(3)*i)/2) = 1,_
a(1) = j_1((-1+sqrt(3)*i)/2) = -744 + 0^1 = -744,
a(2) = j_2((-1+sqrt(3)*i)/2) = 159768 - 1488*0^1 + 0^2 = 159768. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A004009 (E_4), A110163, A013973 (E_6).
E_{k+2}/E_k: A288877 (k=2), this sequence (k=4, 8), A288840 (k=6).
Cf. A000521 (j), A035230 (-q*j'), A066395 (1/j), A289141.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 13; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[6]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
a[ n_] := With[{j = Series[1728 KleinInvariantJ[ Log[ Series[q, {q, 0, n + 1}]]/(2 Pi I)], {q, 0, n}]}, SeriesCoefficient[ -q D[j, q] / j, {q, 0, n}]]; (* Michael Somos, Aug 15 2018 *)

From Seiichi Manyama, Jun 27 2017: (Start)
Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n((-1+sqrt(3)*i)/2).
G.f.: Sum_{n >= 0} j_n((-1+sqrt(3)*i)/2)*q^n. (End)
a(n) ~ (-1)^n * 3 * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jun 28 2017
G.f.: -q*j'/j where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

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

A211342 Decimal expansion of q between 0 and 1 maximizing Dedekind eta function eta(q) = q^(1/24) * Product_{n>=1} (1 - q^n).

Original entry on oeis.org

0, 3, 7, 2, 7, 6, 8, 1, 0, 2, 9, 6, 4, 5, 1, 6, 5, 8, 1, 5, 0, 9, 8, 0, 7, 8, 5, 6, 5, 1, 6, 4, 4, 6, 1, 8, 0, 3, 6, 2, 8, 2, 3, 7, 9, 4, 8, 2, 7, 8, 3, 0, 0, 6, 7, 0, 4, 1, 0, 2, 2, 1, 3, 4, 7, 7, 5, 1, 3, 9, 2, 9, 1, 0, 2, 0, 3, 6, 7, 5, 5, 3, 2, 3, 0, 0, 3, 4, 3, 1, 4, 7, 0, 6, 5, 8, 2, 9, 8, 9, 0

Offset: 0

Jean-François Alcover, Feb 05 2013

			0.0372768102964516581509807856516446180362823794827830067...

Eric Weisstein's MathWorld, Dedekind Eta Function

Cf. A000203, A057823, A288877, A289392.

q /. Last @ FindMaximum[ DedekindEta[ -I*Log[q]/(2*Pi)], {q, 1/25}, WorkingPrecision -> 200] // RealDigits[#][[1]][[1 ;; 100]]& // Prepend[#, 0]&
x /. FindRoot[24*Sum[DivisorSigma[1, k]*x^k, {k, 1, 1000}] == 1, {x, 1}, WorkingPrecision -> 101] (* Vaclav Kotesovec, Jun 28 2017 *)

Root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jun 28 2017

Equals A057823^2. - Vaclav Kotesovec, Jul 02 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

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

Original entry on oeis.org

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).

A294181 Coefficients in expansion of E_2/E_4.

Original entry on oeis.org

1, -264, 61128, -14107296, 3255470952, -751247454384, 173361309784992, -40005651284526912, 9231887649122522280, -2130392752758423726312, 491619206548389935051568, -113448303808924351510423008, 26179851123971817380111236128

Offset: 0

Seiichi Manyama, Feb 11 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A001943, A004009 (E_4), A006352 (E_2), A288877.

E_k/E_{k+2}: this sequence (k=2), A294182 (k=4), A294183 (k=6).

terms = 13;
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}];
E2[x]/E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Convolution inverse of A288877.

a(n) ~ (-1)^n * 1024 * Pi^11 * exp(Pi*sqrt(3)*n) / (3^(3/2) * Gamma(1/3)^18). - Vaclav Kotesovec, Jun 03 2018

cp's OEIS Frontend

A294626 a(n) = (1/(24*n)) * Sum_{d|n} A008683(n/d) * (A288877(d) - A288261(d)).

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A294975 a(n) = (1/(24*n)) * Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

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A288261 Coefficients in expansion of E_6/E_4.

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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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A211342 Decimal expansion of q between 0 and 1 maximizing Dedekind eta function eta(q) = q^(1/24) * Product_{n>=1} (1 - q^n).

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

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

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A294181 Coefficients in expansion of E_2/E_4.

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A294626 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288877(d) - A288261(d)).

A294975 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

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) ... .