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.

Previous Showing 11-17 of 17 results.

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Formula

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

A289397 Coefficients in expansion of (q*j(q))^(-1/24).

Original entry on oeis.org

1, -31, 3809, -620190, 111669570, -21246138749, 4186228503780, -845058129488699, 173647689528542310, -36170751826552656600, 7615730581866678419370, -1617501058117655447210580, 346019784662582818549094159
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2017

Keywords

Links

Crossrefs

(q*j(q))^(k/24): this sequence (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).
Cf. A000521 (j(q)), A066395.

Programs

  • Mathematica
    (q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^(-1/24) + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A192731(n)/24) = Product_{n>=1} (1-q^n)^(1-A289395(n)).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = 0.13397834215417716857261649901051678539339753563926756586381... = 2^(1/8) * exp(Pi/(8 * sqrt(3))) * sqrt(Pi) / (3^(1/8) * Gamma(1/8) * Gamma(1/3)^(3/4)). - Vaclav Kotesovec, Mar 05 2018, updated Mar 06 2018
a(n) * A106205(n) ~ c * exp(2*Pi*sqrt(3)*n) / n^2, where c = -sqrt(2-sqrt(2)) / (16*Pi). - Vaclav Kotesovec, Mar 06 2018

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

Views

Author

Seiichi Manyama, Jun 18 2017

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

  • 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).

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Formula

(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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

Programs

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

Formula

(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

A299827 Coefficients in expansion of (q*j(q))^(-1/8) where j(q) is the elliptic modular invariant (A000521).

Original entry on oeis.org

1, -93, 14310, -2598835, 504870840, -101820075030, 21033065244233, -4418043012449640, 939524696045366400, -201695299876429277490, 43625340820210623183729, -9493467131549164702157730, 2076344691467486382060290550
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2018

Keywords

Crossrefs

Cf. A000521, A289298.

Programs

  • Mathematica
    CoefficientList[Series[(2 * QPochhammer[-1, x])^3 / (65536 + x*QPochhammer[-1, x]^24)^(3/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 20 2018 *)

Formula

Convolution inverse of A289298.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/8), where c = 0.433852674132039602551793002786117867365165961976868338756... = 2^(3/8) * exp(sqrt(3) * Pi/8) * Pi^(3/2) / (3^(3/8) * Gamma(1/3)^(9/4) * Gamma(3/8)). - Vaclav Kotesovec, Feb 20 2018, updated Mar 06 2018
a(n) * A289298(n) ~ -3*2^(1/4)*sqrt(1+sqrt(2)) * exp(2*sqrt(3)*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Feb 20 2018
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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