A289397 -id:A289397 - 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

A028515 Expansion of A007245^6.

Original entry on oeis.org

1, 1488, 947304, 335950912, 72474624276, 9790124955552, 833107628914688, 45630592148400000, 1754954450906393538, 51062104386000089648, 1186840963302480101376, 22924552119951492244800, 378933532779364657975000

Offset: 0

N. J. A. Sloane

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A000521 (j(q)).

(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), A161361 (k=12), A028512 (k=16), A028513 (k=32), A028514 (k=40), this sequence (k=48), A288846 (k=72).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^6, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2017 *)

a(n) ~ exp(4*Pi*sqrt(2*n)) / (2^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 29 2017

(q*j(q))^2 where j(q) is the elliptic modular invariant. - Seiichi Manyama, Jul 15 2017

A288846 Expansion of (q*j)^3, where j is a modular function A000521.

Original entry on oeis.org

1, 2232, 2251260, 1355202240, 541778118390, 151522053809760, 30456116651640888, 4460775211418664960, 479919718908048515625, 38292247221915373896560, 2309356967925215526546564, 108570959012192293978767360, 4111854826236389868361040550

Offset: 0

Seiichi Manyama, Jun 18 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A000521 (j(q)), A004009 (E_4), A008411 (E_4^3).

(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), A161361 (k=12), A028512 (k=16), A028513 (k=32), A028514 (k=40), A028515 (k=48), this sequence (k=72).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^3 + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: ((1 + 240 Sum_{k>0} k^3 q^k/(1-q^k))^3/(Product_{k>0} (1-q^k)^24))^3.

a(n) ~ 3^(1/4) * exp(4*Pi*sqrt(3*n)) / (sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jun 29 2017

A028512 Character of extremal vertex operator algebra of rank 16.

Original entry on oeis.org

1, 496, 69752, 2115008, 34670620, 394460000, 3499148224, 25817318016, 165011628166, 939112182480, 4853601292512, 23116070653888, 102602164703800, 428200065370144, 1692346392263680, 6371305129660032

Offset: 0

N. J. A. Sloane

G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Mathematische Schriften, Vol. 286 (1996), 1-85.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
G. Hoehn (gerald(AT)math.ksu.edu), Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Doctoral Dissertation, Univ. Bonn, Jul 15 1995 (pdf, ps).

Cf. A000521 (j(q)).

(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), A161361 (k=12), this sequence (k=16), A028513 (k=32), A028514 (k=40), A028515 (k=48).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 15 2017 *)
CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^2 / (2*QPochhammer[-1, x])^16, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)

Square of A007245.
(q*j(q))^(2/3) where j(q) is the elliptic modular invariant. - Seiichi Manyama, Jul 15 2017
a(n) ~ exp(4*Pi*sqrt(2*n/3)) / (6^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 15 2017

A028513 Expansion of A007245^4.

Original entry on oeis.org

1, 992, 385520, 73424000, 7032770680, 330234251072, 9708251628992, 205208814844160, 3384709979113500, 45920987396301280, 531402725344000864, 5384625599438260096, 48726640432968418240, 399835655086212744000

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000521 (j(q)).

(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), A161361 (k=12), A028512 (k=16), this sequence (k=32), A028514 (k=40), A028515 (k=48).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 15 2017 *)

(q*j(q))^(4/3) where j(q) is the elliptic modular invariant. - Seiichi Manyama, Jul 15 2017

a(n) ~ exp(8*Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 15 2017

A028514 Expansion of A007245^5.

Original entry on oeis.org

1, 1240, 635660, 173158720, 26866494270, 2390772025248, 123244340937400, 4235204881123840, 107367902876988285, 2147149471392237840, 35461233105160369124, 499800581310885326080, 6159994549959101077830

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000521 (j(q)).

(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), A161361 (k=12), A028512 (k=16), A028513 (k=32), this sequence (k=40), A028515 (k=48).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 15 2017 *)

(q*j(q))^(5/3) where j(q) is the elliptic modular invariant. - Seiichi Manyama, Jul 15 2017

a(n) ~ 5^(1/4) * exp(4*Pi*sqrt(5*n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 15 2017

A305696 Coefficients of (q*(j(q)-744))^(-1/4) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, -49221, -5373440, 5840692110, 1317368987136, -769081921703395, -285861152927176704, 99587019847435059600, 58472021328782000084480, -11456674101843809483255526, -11455351916487867258761894400, 892125673948866841204086469705

Offset: 0

Seiichi Manyama, Jun 08 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..377
Eric Weisstein's World of Mathematics, j-Function

(q*(j(q)-744))^(k/4): A305699 (k=-4), A305698 (k=-2), this sequence (k=-1), A304020 (k=1), A305697 (k=2).

Cf. A000521 (j), A014708 (j-744), A289397, A289416, A302407.

CoefficientList[Series[((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x)^(-1/4), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

G.f.: Product_{k>0} (1 - x^k)^(A302407(k)/4).

A299826 Coefficients in expansion of (q*j(q))^(-1/12) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, -62, 8579, -1476538, 276299401, -54140398258, 10925052030358, -2250028212438240, 470403050272649518, -99482921702360817662, 21231436164082720565341, -4564732260005808181200000, 987422026920066412423809840

Offset: 0

Seiichi Manyama, Feb 20 2018

sign

(q*j(q))^(k/24): A289397 (k=-1), this sequence (k=-2), A299827 (k=-3), A299828 (k=-4), A299829 (k=-5), A299830 (k=-6), A299831 (k=-8), A299832 (k=-12).

Cf. A000521, A289297.

CoefficientList[Series[(2 * QPochhammer[-1, x])^2 / (65536 + x*QPochhammer[-1, x]^24)^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 20 2018 *)

Convolution inverse of A289297.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(3/4), where c = 0.28101701912289268934379724324854717406285519051128823261445... = 2^(1/4) * exp(Pi/(4 * sqrt(3))) * Pi / (3^(1/4) * Gamma(1/4) * Gamma(1/3)^(3/2)). - Vaclav Kotesovec, Feb 20 2018, updated Mar 06 2018
a(n) * A289297(n) ~ -exp(2*sqrt(3)*n*Pi) / (2^(5/2)*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018

cp's OEIS Frontend

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

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A161361 Convolution square root of A000521.

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A028515 Expansion of A007245^6.

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A288846 Expansion of (q*j)^3, where j is a modular function A000521.

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A028512 Character of extremal vertex operator algebra of rank 16.

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A028513 Expansion of A007245^4.

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A028514 Expansion of A007245^5.

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A305696 Coefficients of (q*(j(q)-744))^(-1/4) where j(q) is the elliptic modular invariant.

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