cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

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

Original entry on oeis.org

1, 31, -2848, 413823, -68767135, 12310047967, -2309368876639, 447436508910495, -88755684988520798, 17924937024841839390, -3671642907594608226078, 760722183234128461061246, -159105706560247952472114973
Offset: 0

Views

Author

Michael Somos, Apr 25 2005

Keywords

Comments

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)

Examples

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

Links

Crossrefs

(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.

Programs

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

Formula

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

Views

Author

Gary W. Adamson, Jun 07 2009

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

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).

Programs

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

Formula

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

Extensions

More terms from R. J. Mathar, Jun 15 2009
Keyword:sign introduced by R. J. Mathar, Jul 07 2009

A289297 Expansion of (q*j(q))^(1/12) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, 62, -4735, 651070, -103766140, 17999397756, -3292567703035, 624659270035130, -121698860487451255, 24194029851560118900, -4886913657541566648179, 999849040331683393909232, -206741394604073327046805355
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(q*j(q))^(k/24): A106205 (k=1), this sequence (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.

Programs

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

Formula

G.f.: Product_{n>=1} (1-q^n)^(A192731(n)/12).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = 0.200236163401945306105645017761063156355568043417672219092096121424... = 3^(1/4) * Gamma(1/4) * Gamma(1/3)^(3/2) / (2^(11/4) * exp(Pi/(4 * sqrt(3))) * Pi^2). - Vaclav Kotesovec, Jul 03 2017, updated Mar 06 2018
a(n) * A299826(n) ~ -exp(2*sqrt(3)*n*Pi) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

(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.

Programs

  • Mathematica
    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 *)

Formula

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
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