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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A337445 A291159 A104516 * A158503 A179506 A185653
KEYWORD
sign
AUTHOR
Lynette O'Brien, Dec 15 2016
STATUS
approved

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Last modified May 23 10:34 EDT 2024. Contains 372760 sequences. (Running on oeis4.)