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.

A030201 Expansion of eta(q^3)*eta(q^21).

Original entry on oeis.org

0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Multiplicative. See A002655 for formula. - Andrew Howroyd, Aug 05 2018

Links

Crossrefs

Expansion of eta(q^k)*eta(q^(24 - k)): A030199 (k=1), this sequence (k=3), A030213 (k=5), A030214 (k=7), A030215 (k=9), A030216 (k=10), A030217 (k=11).
Cf. A002655.

Programs

  • Mathematica
    q QPochhammer[q^3] QPochhammer[q^21] + O[q]^105 // CoefficientList[#, q]& (* Jean-François Alcover, Sep 06 2019 *)
  • PARI
    seq(n)={concat([0], Vec(eta(x^3 + O(x*x^n)) * eta(x^21 + O(x*x^n))))} \\ Andrew Howroyd, Aug 05 2018

Formula

Expansion of x * Product_{k>=1} (1 - x^(3*k)) * (1 - x^(21*k)). - Seiichi Manyama, Oct 18 2016
a(3*n + 1) = A002655(n), a(3*n) = a(3*n + 2) = 0. - Andrew Howroyd, Aug 05 2018

A160806 Expansion of q^(-1/3) * (eta(q) * eta(q^7) + eta(q^4) * eta(q^28)) in powers of q^2.

Original entry on oeis.org

1, -1, 0, 0, 1, 0, -2, -2, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, -2, 0, -1, 2, 0, 0, 0, -2, 0, 2, 1, -1, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, -2, -2, -1, 0, 0, 1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0, -2, 0, -2, 0, 0, 0, 0, -1, 0, 2, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1
Offset: 0

Views

Author

Michael Somos, May 26 2009

Keywords

Examples

			G.f. = q - q^7 + q^25 - 2*q^37 - 2*q^43 + q^49 + 2*q^67 + 2*q^79 - 2*q^109 + ...

Links

Programs

  • Mathematica
    eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/6)*(eta[q^(1/2)]*eta[q^(7/2)] + eta[q^2]*eta[q^14]), {q, 0, 100}], q] (* G. C. Greubel, Sep 21 2018 *)
  • PARI
    {a(n) = local(A, p, e, x, y); if(n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==7, (-1)^e, if(kronecker(p,7)==-1, !(e%2), for(x=0,sqrtint(p\7), if(issquare(p - 7*x^2, &y), y=if(x%3&y%3, real(I^e), (e+1) * if(x%3, (-1)^e, 1)); break)); y)))))}
  • PARI
    {a(n) = local(A); if(n<0, 0, n *= 2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A), n))}

Formula

a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-1)^e if p = 7, b(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7), else p == 1, 2, 4 (mod 7) and p=y^2+7x^2 when b(p^2e) = (-1)^e if x*y not divisible by 3, b(p^e) = e+1 if x divisible by 3 or (e+1)(-1)^e if y divisible by 3.
A002655(2*n) = a(n).

A058565 McKay-Thompson series of class 21C for the Monster group.

Original entry on oeis.org

1, 3, 8, 11, 25, 35, 57, 86, 139, 198, 291, 417, 588, 812, 1132, 1538, 2103, 2805, 3767, 4963, 6554, 8548, 11165, 14426, 18601, 23830, 30443, 38642, 48986, 61748, 77669, 97206, 121478, 151067, 187556, 231974, 286385, 352340, 432641, 529688, 647241, 788738, 959470, 1164291, 1410386
Offset: 0

Views

Author

N. J. A. Sloane, Nov 27 2000

Keywords

Examples

			G.f. = 1 + 3*x + 8*x^2 + 11*x^3 + 25*x^4 + 35*x^5 + 57*x^6 + 86*x^7 + ... -  _Michael Somos_, Feb 26 2017
T21C = 1/q + 3*q^2 + 8*q^5 + 11*q^8 + 25*q^11 + 35*q^14 + 57*q^17 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 176 Entry 32(iii).

Links

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
Cf. A002652, A002655, A282877.

Programs

  • Mathematica
    a[ n_] := With[ {A = (QPochhammer[ x^7] / QPochhammer[ x])^4}, SeriesCoefficient[ (1/A + 13 x + 49 x^2 A)^(1/3), {x, 0, n}]]; (*  Michael Somos, Feb 26 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(1/3)*(eta[q]*eta[q^7]/(eta[q^2] *eta[q^14])); a:= CoefficientList[Series[(A + 4*q/A^2), {q,0,60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 21 2018 *)
a[ n_] := With[ {A1 = QPochhammer[ x] QPochhammer[ x^7], A2 = QPochhammer[ x^2] QPochhammer[ x^14]}, SeriesCoefficient[ (A1^3 + 4 x A2^3) / (A1^2 A2), {x, 0, n}]]; (*  Michael Somos, Oct 27 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1/A + 13*x + 49*x^2 * A)^(1/3), n))}; /*  Michael Somos, Feb 26 2017 */
  • PARI
    q='q+O('q^50); A = (eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))); Vec(A + 4*q/A^2) \\ G. C. Greubel, Jun 21 2018
  • PARI
    {a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1^2 * A2), n))}; /* Michael Somos, Oct 27 2018 */

Formula

From Michael Somos, Feb 26 2017: (Start)
Expansion of f(-x^7, -x^14)^2 / f(-x, -x^2) * (w3/w1^2 + x*w2/w3^2 - x*w1/w2^2) in powers of x where w1 = f(-x, -x^6), w2 = f(-x^2, -x^5), w3 = f(-x^3, -x^4) and f(, ) is Ramanujan's general theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = f(t) where q = exp(2 Pi i t).
Convolution cube is A282877.
Convolution product with A002655 is A002652. (End)
Expansion of A + 4*q/A^2, where A = q^(1/3)*(eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))), in powers of q. - G. C. Greubel, Jun 21 2018
a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Feb 26 2017

Extensions

Terms a(8) onward added by G. C. Greubel, Jun 21 2018
Showing 1-3 of 3 results.

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

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