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A243747
Expansion of (phi(q) - phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.
2
1, -2, 1, 2, -2, 0, 1, -2, 4, -2, -2, 2, 0, 0, 1, 0, -1, -2, 4, 0, -2, 0, -2, 2, 4, -4, 0, 2, 0, 0, 1, -4, 2, 0, -1, 2, -2, 0, 4, 0, 0, -2, -2, 2, 0, 0, -2, 0, 5, -4, 4, 2, -4, 0, 0, -4, 4, -2, 0, 2, 0, 0, 1, 4, -4, -2, 2, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0
OFFSET
2,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Expansion of third basis element of modular forms space for Gamma_1(8) of weight 1 in powers of q.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (q * f(-q, -q^7)^2 / psi(-q))^2 in powers of q where psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [-2, 0, 2, 2, 2, 0, -2, -2, ...].
G.f.: (theta_3(x) - theta_3(x^2))^2 / 4 = (Sum_{k>0} x^(k^2) - x^(2k^2))^2.
Convolution square of A143259.
EXAMPLE
G.f. = q^2 - 2*q^3 + q^4 + 2*q^5 - 2*q^6 + q^8 - 2*q^9 + 4*q^10 - 2*q^11 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];
PROG
(PARI) {a(n) = if( n<2, 0, sum(k=1, n-1, (issquare(k) - issquare(2*k)) * (issquare(n - k) - issquare(2*n - 2*k))))};
(Sage) ModularForms( Gamma1(8), 1, prec=70).2
(Magma) Basis( ModularForms( Gamma1(8), 1), 70) [3];
CROSSREFS
CF. A143259.
Sequence in context: A071858 A122864 A140084 * A105937 A035146 A035216
KEYWORD
sign
AUTHOR
Michael Somos, Jun 09 2014
STATUS
approved