OFFSET
0,2
COMMENTS
LINKS
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 16 sequence [2, -3, 2, 3, 2, -3, 2, -7, 2, -3, 2, 3, 2, -3, 2, -3, ...].
Expansion of phi(x) * phi(x^4)^2 = phi(x^4)^3 + 2*x*phi(x^4)*psi(x^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
G.f.: theta_3(q) * theta_3(q^4)^2, where theta_3() is the Jacobi theta function.
EXAMPLE
G.f. = 1 + 2*x + 6*x^4 + 8*x^5 + 12*x^8 + 10*x^9 + 8*x^12 + 8*x^13 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^4]^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^10 / (eta(x + A)^2 * eta(x^4 + A)^6 * eta(x^16 + A)^4), n))};
(Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 77); A[1] + 2*A[2];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 17 2019
STATUS
approved