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A145782 Expansion of (chi(q^3) * chi(q^5))^2 / (chi(q) * chi(q^15)) in powers of q where chi() is a Ramanujan theta function. 2
1, -1, 1, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, -1, 2, -1, -2, 3, -1, -1, 2, -3, 0, 3, -1, -2, 2, 0, -2, 6, -3, -4, 7, -3, -2, 5, -6, -2, 8, -3, -5, 6, -2, -4, 12, -7, -10, 15, -6, -5, 13, -12, -4, 18, -7, -11, 14, -6, -10, 24, -14, -20, 32, -12, -12, 29, -24, -9 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,15

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^10)^4 * eta(q^15) * eta(q^60) / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30))^2 in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

a(n) = - A145726(n) unless n=0. Convolution inverse of A145783.

a(2*n) = A094022(n) unless n=0. - Michael Somos, Sep 04 2015

EXAMPLE

G.f. = 1 - q + q^2 + q^5 - q^6 + q^8 - q^10 - q^13 + 2*q^14 - q^15 - 2*q^16 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[  (QPochhammer[ -x^3, x^6] QPochhammer[ -x^5, x^10] )^2 / (QPochhammer[ -x, x^2] QPochhammer[ -x^15, x^30]), {x, 0, n}]; (* Michael Somos, Sep 04 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^10 + A)^4 * eta(x^15 + A) * eta(x^60 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A))^2, n))};

(MAGMA) S<x> := PowerSeriesRing(RationalField()); Coefficients( DedekindEta(x)*DedekindEta(x^4) *DedekindEta(x^6)^4*DedekindEta(x^10)^4* DedekindEta(x^15)*DedekindEta(x^60)/(DedekindEta(x^2)*DedekindEta(x^3) *DedekindEta(x^5)*DedekindEta(x^12)*DedekindEta(x^20)*DedekindEta(x^30) )^2); // G. C. Greubel, Mar 04 2018

CROSSREFS

Cf. A094022, A145726, A145783.

Sequence in context: A273132 A294859 A336320 * A131797 A145727 A131796

Adjacent sequences:  A145779 A145780 A145781 * A145783 A145784 A145785

KEYWORD

sign

AUTHOR

Michael Somos, Oct 23 2008

STATUS

approved

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Last modified June 13 15:33 EDT 2021. Contains 345008 sequences. (Running on oeis4.)