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 1-3 of 3 results.

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

A299413 Coefficients in expansion of (E_6^2/E_4^3)^(1/2).

Original entry on oeis.org

1, -864, 269568, -75240576, 19930724352, -5124295980864, 1292387210099712, -321604751662509312, 79241739168490536960, -19376923061550541800672, 4709786462808256974509568, -1139188440993923671697455488
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2018

Keywords

Links

Crossrefs

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), this sequence (k=144), A289210 (k=288).
Cf. A000521 (j), A007242, A299832.

Programs

  • Mathematica
    terms = 12;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}] + O[x]^terms; (E6[x]^2/E4[x]^3)^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 22 2018 *)

Formula

G.f.: (1 - 1728/j)^(1/2), where j is the j-function.
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * sqrt(n), where c = 32*sqrt(2) * Pi^(11/2) / Gamma(1/3)^9. - Vaclav Kotesovec, Mar 04 2018

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

Views

Author

Seiichi Manyama, Feb 20 2018

Keywords

Crossrefs

(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.

Programs

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

Formula

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
Showing 1-3 of 3 results.

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