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
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
KEYWORD
sign
AUTHOR
Lynette O'Brien, Dec 15 2016
STATUS
approved