OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-7/6) * eta(q^2) * eta(q^6)^5 / (eta(q) * eta(q^3)) in powers of q.
Expansion of b(x^2) * c(x^2)^2 / sqrt(b(x) * c(x)/3) in powers of x where b(), c() are cubic AGM theta functions.
Expansion of psi(x^3) * f(-x^6)^3 / chi(-x) = f(-x^6)^4 / (chi(-x) * chi(-x^3)) = f(-x^6)^5 /(f(-x^3) * chi(-x)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 - x^(6*k))^4.
EXAMPLE
G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + ...
G.f. = q^7 + q^13 + q^19 + 3*q^25 + 3*q^31 + 4*q^37 + 3*q^43 + 4*q^49 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^4 / (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^5 / (QPochhammer[ x, x^2] QPochhammer[ x^3]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^6]^5 / (QPochhammer[ x] QPochhammer[x^3]), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)), n))};
(Magma) Basis( ModularForms( Gamma0(36), 2), 427) [8];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 09 2018
STATUS
approved