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 1-2 of 2 results.

A279613 Expansion of the g.f. of A160534 in powers of A121593.

Original entry on oeis.org

1, -7, 42, -231, 1155, -4998, 15827, -791, -566244, 6506955, -53524611, 369879930, -2218053747, 11306008875, -43772711220, 55203364377, 1172838094533, -16542312772356, 150992704165079, -1130142960861845, 7290759457923816
Offset: 1

Views

Author

Lynette O'Brien, Dec 15 2016

Keywords

Comments

(eta(q))^7/eta(7*q) in powers of (eta(7*q)/eta(q))^4.
This sequence is u_n in Theorem 6.5 in O'Brien's thesis.

Examples

			G.f.: 1 - 7*x + 42*x^2 - 231*x^3 + 1155*x^4 - 4998*x^5 + ...

References

  • L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

Links

Crossrefs

Cf. A183204, A229111, A279618, A279619.

Formula

(n+1)^4a_7(n+1)=-(26*n^4+52*n^3+58*n^2+32*n+7)a_7(n)-(267*n^4+268*n^2+18)a_7(n-1)-(1274*n^4-2548*n^3+2842*n^2-1568*n+343)a_7(n-2)-2401(n-1)^4a_7(n-3)
with a_7(0)=1, a_7(-1)=a_7(-2)=a_7(-3)=0.
asymptotic conjecture: a(n) ~ C n^(-4/3) 7^n cos( n( arctan( (3*sqrt 3)/13) +Pi -1.083913253)), where C = 6.502807770...

A279618 Expansion of w_7/(1 + 13*w_7 + 49*w_7^2) in powers of q, where w_7 = (eta(7*q)/eta(q))^4.

Original entry on oeis.org

1, -9, 30, -15, -240, 978, -1463, -2361, 18201, -42800, 15624, 227742, -809028, 1088367, 1593120, -11383551, 25003158, -8589729, -119069358, 403991280, -521730930, -736063496, 5088063696, -10843708302, 3624181875, 48991048836, -162420646812, 205328313785, 284014016994
Offset: 1

Views

Author

Lynette O'Brien, Dec 15 2016

Keywords

Comments

G.f. is y_7 in Cooper's paper.
See Equation (3.15) and Theorem 3.10 in O'Brien's thesis.
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 07 2018

Examples

			G.f. = q - 9*q^2 + 30*q^3 - 15*q^4 - 240*q^5 + 978*q^6 - 1463*q^7 + ...

References

  • S. Cooper, (2012). Sporadic sequences, modular forms and new series for 1/pi. The Ramanujan Journal, 29(1-3), 163-183.
  • L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

Links

Crossrefs

Cf. A002652, A121593, A279613, A279619.

Programs

  • Mathematica
    a[ n_] := With[{u1 = QPochhammer[ x]^4, u7 = QPochhammer[ x^7]^4}, SeriesCoefficient[ x u1 u7 / (u1^2 + 13 x u1 u7 + 49 x^2 u7^2) , {x, 0, n}]]; (* Michael Somos, Sep 07 2018 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, A = x * O(x^n); A = x * (eta(x^7 + A) / eta(x + A))^4; polcoeff( 1 / (1/A + 13 + 49*A), n))}; /* Michael Somos, Sep 07 2018 */

Formula

G.f. is w_7/(1 + 13*w_7 + 49*w_7^2) = (eta(q)*eta(7q)/z_7)^3 where w_7 = (eta(7*q)/eta(q))^4 and z_7 = 1 + 2*Sum_{k>0} Kronecker(-7,k)*q^k/(1-q^k).
G.f. is also (eta(q)*eta(7*q)/z_7)^3, where z_7 = 1 + 2*Sum_{k>0} Kronecker(-7,k)*q^k/(1-q^k). See A002652.
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.