A289297 -id:A289297 - cp's OEIS frontend

A106205 Expansion of (q*j(q))^(1/24) where j(q) is the elliptic modular invariant (A000521).

1, 31, -2848, 413823, -68767135, 12310047967, -2309368876639, 447436508910495, -88755684988520798, 17924937024841839390, -3671642907594608226078, 760722183234128461061246, -159105706560247952472114973

Offset: 0

Michael Somos, Apr 25 2005

sign

From Vaclav Kotesovec, Jun 10 2018: (Start)
For k > 0, if mod(k,8) <> 0 then (q*j(q))^(k/24) is asymptotic to -(-1)^n * sin(k*Pi/8) * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * Gamma(k/8) * exp(Pi*sqrt(3)*n) / (Pi^(k/2 + 1) * 2^(k/8 + 3) * exp(k*Pi/(8*sqrt(3))) * n^(k/8 + 1)). Equivalently, is asymptotic to -(-1)^n * k * 3^(k/8) * Gamma(1/3)^(3*k/4) * exp(Pi*sqrt(3)*(n - k/24)) / (Pi^(k/2) * 2^(k/8 + 3) * Gamma(1 - k/8) * n^(k/8 + 1)).
For k > 0, if mod(k,8) = 0 then (q*j(q))^(k/24) is asymptotic to exp(Pi*sqrt(2*k*n/3)) * k^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).
(End)

			1 + 31*q - 2848*q^2 + 413823*q^3 - 68767135*q^4 + 12310047967*q^5 - 2309368876639*q^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..424

(q*j(q))^(k/24): this sequence (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(1/8) / (2*QPochhammer[-1, x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

{a(n)=if(n<0,0, polcoeff( (ellj(x+x^2*O(x^n))*x)^(1/24),n))}

This is essentially the eighth root of the theta series of E_8 (A108091), divided by the Dedekind eta function. - N. J. A. Sloane, Aug 08 2005
G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/24). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 0.11364889078525240958152388212499254894082832445224690827436413842337... = 3^(1/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(3/4) / (2^(33/8) * exp(Pi/(8 * sqrt(3))) * Pi^(3/2)). - Vaclav Kotesovec, Jul 02 2017, updated Mar 06 2018
a(n) * A289397(n) ~ c * exp(2*Pi*sqrt(3)*n) / n^2, where c = -sqrt(2-sqrt(2)) / (16*Pi). - Vaclav Kotesovec, Mar 06 2018

A161361 Convolution square root of A000521.

Original entry on oeis.org

1, 372, 29250, -134120, 54261375, -6139293372, 854279148734, -128813964933000, 20657907916144515, -3469030105750871000, 603760629237519966018, -108124880417607682194048, 19820541224206810447813500

Offset: 0

Gary W. Adamson, Jun 07 2009

sign

Triangle A161362 = the corresponding convolution triangle with row sums = A000521.

			a(2) = 29250 = 1/2 * (A000521(2) - 372^2) = 1/2 * (196884 - 138384) = 29250.
G.f. = 1 + 372*x + 29250*x^2 - 134120*x^3 + 54261375*x^4 - ...
G.f. = 1/q + 372*q + 29250*q^3 - 134120*q^5 + 54261375*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..426

Cf. A000521, A161362, A192731.

(q*j(q))^(k/24): A289397 (k=-1), A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304(k=10), A289305 (k=11), this sequence (k=12).

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/2) / (4096 * QPochhammer[-1, x]^12), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( sqrt(x * (1 + 256*A)^3 / A), n))}; /* Michael Somos, May 03 2014 */

Given A000521: (j = 1/q + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...); multiply by q and take the convolution square root.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, May 03 2014
G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/2). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 0.378271951998085144930610869223050101960774818... = 3^(5/2) * Gamma(1/3)^9 / (2^(7/2) * exp(sqrt(3) * Pi/2) * Pi^(13/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299832(n) ~ 3*exp(2*sqrt(3)*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

More terms from R. J. Mathar, Jun 15 2009

Keyword:sign introduced by R. J. Mathar, Jul 07 2009

A289298 Expansion of (q*j(q))^(1/8) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 93, -5661, 741532, -113207799, 19015433748, -3390166183729, 629581913929419, -120437982238038210, 23564574046009042869, -4692899968498921291530, 948024211601180444075739, -193775768073341380441728322

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..424

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), this sequence (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/8) / (2*QPochhammer[-1, x])^3, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/8) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/8).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(11/8), where c = 0.2541876595230750963327533839122695596555059904123327336821622582369... = 3^(11/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(9/4) * Gamma(3/8) / (2^(35/8) * exp(sqrt(3) * Pi/8) * Pi^(5/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299827(n) ~ -3*2^(1/4)*sqrt(1+sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

A289299 Expansion of (q*j(q))^(1/6) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 124, -5626, 715000, -104379375, 16966161252, -2946652593626, 535467806605000, -100554207738307500, 19359037551684042500, -3800593180746056684372, 757968936254309704500248, -153133996443087103652605627

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), this sequence (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[Sqrt[65536 + x*QPochhammer[-1, x]^24] / (2*QPochhammer[-1, x])^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/6) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/6).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 0.27174882346571745439868471841345665496773077910099184617347055088... = sqrt(3) * Gamma(1/3)^3 / (2^(3/2) * exp(Pi/(2 * sqrt(3))) * Pi^(5/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299828(n) ~ -exp(2*sqrt(3)*Pi*n) / (2*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 155, -4630, 601265, -83644610, 13148835656, -2223584717035, 395257299676190, -72843145114522035, 13796578308407774725, -2669652272250261922223, 525556527400692937755655, -104937908072571416700653120

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), this sequence (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(5/8) / (2*QPochhammer[-1, x])^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(5/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A192731(n)/24).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(13/8), where c = 0.251632947646443757912747944865268710111059274679945447776728146817... = 5 * 3^(5/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(15/4) * Gamma(5/8) / (2^(37/8) * exp(5 * Pi / (8 * sqrt(3))) * Pi^(7/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299829(n) ~ -5*sqrt(2 + sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

A289301 Expansion of (q*j(q))^(1/4) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 186, -2673, 430118, -56443725, 8578591578, -1411853283028, 245405765574252, -44373155962556475, 8266332741845429800, -1576306833508315403544, 306275559567641721838494, -60432437032381794135586069

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), this sequence (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(3/4) / (64 * QPochhammer[-1, x]^6), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(1/4) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/4).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(7/4), where c = 0.1955865990744763088634116856422381013939034554805874572099292810179... = 3^(7/4) * Gamma(1/3)^(9/2) / (2^(11/4) * exp(sqrt(3) * Pi/4) * Pi^3 * Gamma(1/4)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299830(n) ~ -3*exp(2*sqrt(3)*Pi*n) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

A289302 Expansion of (q*j(q))^(7/24) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 217, 245, 231350, -27293420, 4017072017, -643057897118, 109259930443485, -19377905432572925, 3549922504344871655, -666990037937425724641, 127890778891452935279096, -24934077008209243436961385

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), this sequence (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(7/8) / (2*QPochhammer[-1, x])^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(7/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(7*A192731(n)/24).

a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(15/8), where c = 0.108789720644871714449969800661839212719879897088563371823367481878... = 7 * 3^(7/8) * sqrt(2 - sqrt(2)) * Gamma(1/3)^(21/4) * Gamma(7/8) / (2^(39/8) * exp(7 * Pi / (8 * sqrt(3))) * Pi^(9/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

A289303 Expansion of (q*j(q))^(3/8) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 279, 8964, -129885, 23406255, -3128904747, 473738861853, -76824787699971, 13098300010462845, -2318947179364181165, 422782870045511526012, -78914282330756685655485, 15016013710284896513279286

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), this sequence (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(9/8) / (2*QPochhammer[-1, x])^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(3/8) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(3*A192731(n)/8).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(17/8), where c = 0.1186486859763112993214522284920488979797011156387080809639905476634... = 3^(25/8) * sqrt(2 - sqrt(2)) * Gamma(1/8) * Gamma(1/3)^(27/4) / (2^(65/8) * exp(3 * sqrt(3) * Pi/8) * Pi^(11/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

A289304 Expansion of (q*j(q))^(5/12) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 310, 14765, -232770, 40539830, -5199871688, 765038308115, -121140033966330, 20242157273780710, -3521886754264327670, 632344647471171938140, -116428917411726531951590, 21883035176258955622401245

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), this sequence (k=10), A289305 (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(5/4) / (2*QPochhammer[-1, x])^10, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(5/12) + O[q]^13 //
CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(5*A192731(n)/12).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(9/4), where c = 0.232272469556851820006346410170543574844213494230850435863953522617... = 5 * 3^(5/4) * Gamma(1/4) * Gamma(1/3)^(15/2) / (2^(23/4) * exp(5 * Pi / (4 * sqrt(3))) * Pi^6). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

A289305 Expansion of (q*j(q))^(11/24) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 341, 21527, -244112, 50791235, -6177875286, 883458515093, -136541356378141, 22354744100161913, -3821528558157433970, 675604462786881129711, -122689458583136157060647, 22774615293799045532223797

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

In general, the expansion of (q*j(q))^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 3*m is the positive integer, else is asymptotic to (-1)^n * 3^(3*m) * Gamma(1/3)^(18*m) * exp(Pi*sqrt(3)*(n-m)) / (2^(3*m) * Pi^(12*m) * Gamma(-3*m) * n^(3*m + 1)). - Vaclav Kotesovec, Mar 06 2018

Seiichi Manyama, Table of n, a(n) for n = 0..426

(q*j(q))^(k/24): A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), this sequence (k=11), A161361 (k=12).

Cf. A000521, A192731.

CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^(11/8) / (2*QPochhammer[-1, x])^11, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(11/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(11*A192731(n)/24).

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(19/8), where c = 0.3243413869190225417519777558769755719610962636852073820509897134587... = 11 * 3^(19/8) * sqrt(2 + sqrt(2)) * Gamma(1/3)^(33/4) * Gamma(3/8) / (2^(67/8) * exp(11 * Pi / (8 * sqrt(3))) * Pi^(13/2)). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018

cp's OEIS Frontend

A106205 Expansion of (q*j(q))^(1/24) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A161361 Convolution square root of A000521.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A289298 Expansion of (q*j(q))^(1/8) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289299 Expansion of (q*j(q))^(1/6) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289300 Expansion of (q*j(q))^(5/24) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289301 Expansion of (q*j(q))^(1/4) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289302 Expansion of (q*j(q))^(7/24) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289303 Expansion of (q*j(q))^(3/8) where j(q) is the elliptic modular invariant (A000521).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula