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.

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

1,2

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

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

Table of n, a(n) for n=1..29.

Lynette O'Brien, Modular forms and two new integer sequences at level 7

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

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.

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A002652, A121593, A279613, A279619.

Sequence in context: A337445 A291159 A104516 * A158503 A179506 A185653

Adjacent sequences: A279615 A279616 A279617 * A279619 A279620 A279621

Keyword

sign

Author

Lynette O'Brien, Dec 15 2016

Status

approved