A289061 -id:A289061 - cp's OEIS frontend

A289367 a(n) = (2A288851(n) - 3A110163(n))/288.

6, 717, 398086, 135369240, 62518201350, 27027759382861, 12577742936206854, 5858597459401083456, 2795780972964509144838, 1345924404035022245534925, 655521004499800309096497414, 321708126100955273726273728024

Offset: 1

Seiichi Manyama, Jul 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..368

Cf. A110163, A192731, A288851, A289061, A289365, A289366.

a(n) = (A289061(n) - A192731(n))/288. - Seiichi Manyama, Feb 17 2018

a(n) ~ exp(2*Pi*n) / (144*n). - Vaclav Kotesovec, Jun 03 2018

A289561 Coefficients of 1/(q*(j(q)-1728))^2 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 1968, 2511000, 2605664960, 2387651205420, 2011663789279200, 1594903822090229312, 1207416525204065938560, 881461062200198781904590, 624887481909094711741279120, 432393768184906363401468637728, 293171504960988659691658645670592

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..363

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), this sequence (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-2*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^3, where c = Gamma(3/4)^16 * exp(4*Pi) / (629856 * Pi^4) = 0.120838515551739021017044909469013807578104459775498957232984908667972... - Vaclav Kotesovec, Mar 07 2018

A289562 Coefficients of 1/(q*(j(q)-1728))^3 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 2952, 5218884, 7138351488, 8319960432666, 8678332561127616, 8338315178481134040, 7518590274496806176256, 6444205834302869333758299, 5298802621872639665867604832, 4208666443076672300677008045636, 3246069554930472099322915758511872

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..362

(q*(j(q)-1728))^(k/24): A289563 (k=-96), this sequence (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-3*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^5, where c = Gamma(3/4)^24 * exp(6*Pi) / (4081466880 * Pi^6) = 0.0051446247390864841578336638645072392120317488530740050289688... - Vaclav Kotesovec, Mar 07 2018

A289563 Coefficients of 1/(q*(j(q)-1728))^4 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 3936, 8895024, 15094625920, 21336320693400, 26506772152211520, 29887990556174431424, 31237788209244729015552, 30709242534935581933885740, 28700724444538653431660487520, 25706227251014342788669659769056, 22202613798662970539127791744222592

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..361

(q*(j(q)-1728))^(k/24): this sequence (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-4*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^7, where c = Gamma(3/4)^32 * exp(8*Pi) / (55540601303040 * Pi^8) = 0.0001042996202910562374208781457852661312263780276025385904... - Vaclav Kotesovec, Mar 07 2018

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).

A305757 Inverse Euler transform of q*(j-720) where j is j-function (A000521).

Original entry on oeis.org

24, 196584, 16773144, -18919981056, -3292295086056, 2312547886368744, 640457437563740184, -302667453389051314176, -123005476312830648176616, 39529719620247267255853032, 23306082528463942764630528024, -4849033309391159571741461446656

Offset: 1

Seiichi Manyama, Jun 10 2018

sign

(Conjecture) Let {b_n} = inverse Euler transform of (q*(j+144*k)). b_n is a multiple of 24.

			(1-x)^(-24) * (1-x^2)^(-196584) * (1-x^3)^(-16773144) * (1-x^4)^18919981056 * ... = 1 + 24*x + 196884*x^2 + 21493760*x^3 + 864299970*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..377
N. J. A. Sloane, Transforms

Inverse Euler transform of q*(j+144*k): (-1)*A192731 (k=0), this sequence (k=-5), (-1)*A289061 (k=-12).

Cf. A000521, A007240 (j-720), A302407, A305756.

q*(j-720) = Product_{k>0} (1 - x^k)^(-a(k)).

A289340 Coefficients of (q*(j(q)-1728))^(1/3) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -328, -41956, -8596032, -2597408634, -916285828640, -352170121921992, -143129703441671168, -60517599938503137519, -26355020095077489965264, -11743692598044815023990588, -5329748160859504303225598464

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

In general, the expansion of (q*(j(q)-1728))^m, where j(q) is the elliptic modular invariant (A000521), and m <> 0, is asymptotic to exp(4*Pi*sqrt(m*n)) * m^(1/4) / (sqrt(2) * n^(3/4)) if 2*m is the positive integer, else is asymptotic to 2^(2*m) * 3^(4*m) * Pi^(2*m) * exp(2*Pi*(n-m)) / (Gamma(-2*m) * Gamma(3/4)^(8*m) * n^(2*m + 1)). - Vaclav Kotesovec, Mar 07 2018

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A289339 (k=7), this sequence (k=8), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/3).

a(n) ~ c * exp(2*Pi*n) / n^(5/3), where c = -2^(2/3) * 3^(5/6) * exp(-2*Pi/3) * Gamma(2/3) / (Pi^(1/3) * Gamma(3/4)^(8/3)) = -0.262554753987597280323546158564... - Vaclav Kotesovec, Mar 07 2018

A289339 Coefficients of (q*(j(q)-1728))^(7/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -287, -42595, -9750370, -3081185660, -1117168154431, -438204467218406, -181018051263504195, -77584080248087108885, -34183723168674046275385, -15388633770558568711781905, -7047808475666778827478858184

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

(q*(j(q)-1728))^(k/24): A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), this sequence (k=7), A289340 (k=8), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(7/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{k>=1} (1-q^k)^(7*A289061(k)/24).

a(n) ~ c * exp(2*Pi*n) / n^(19/12), where c = -7 * exp(-7*Pi/12) * Gamma(1/12) / (2^(35/12) * 3^(1/12) * Pi^(17/12) * Gamma(3/4)^(1/3)) = -0.287342744567300675294730727139553541489784437990631575713791583301655... - Vaclav Kotesovec, Mar 07 2018

cp's OEIS Frontend

A289367 a(n) = (2*A288851(n) - 3*A110163(n))/288.

Views

Author

Keywords

Links

Crossrefs

Formula

A289561 Coefficients of 1/(q*(j(q)-1728))^2 where j(q) is the elliptic modular invariant.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289562 Coefficients of 1/(q*(j(q)-1728))^3 where j(q) is the elliptic modular invariant.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289563 Coefficients of 1/(q*(j(q)-1728))^4 where j(q) is the elliptic modular invariant.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Formula

A305757 Inverse Euler transform of q*(j-720) where j is j-function (A000521).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A289340 Coefficients of (q*(j(q)-1728))^(1/3) where j(q) is the elliptic modular invariant.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A289339 Coefficients of (q*(j(q)-1728))^(7/24) where j(q) is the elliptic modular invariant.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A289367 a(n) = (2A288851(n) - 3A110163(n))/288.