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A230205
Expansion of phi(-x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.
1
1, -1, -2, 0, 2, 2, 0, 1, -2, -2, -1, 0, 0, 0, 2, 0, 0, 2, 0, -2, 0, 0, 1, 0, 0, -2, 2, 1, -2, 0, 0, 0, -2, 0, 0, -2, 0, 2, 2, 0, 0, 0, 0, 4, 0, 1, 0, -2, 0, 0, -2, 0, -1, -2, -2, 0, 0, 0, 2, -2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, -2, 0, 0, 0, 2, 0, -1, -4, 0, 0
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 16 sequence [ -1, -2, -2, -1, -2, -1, -1, -2, -1, -1, -2, -1, -2, -2, -1, -2, ...].
a(n) = - A030204(2*n + 1).
EXAMPLE
G.f. = 1 - x - 2*x^2 + 2*x^4 + 2*x^5 + x^7 - 2*x^8 - 2*x^9 - x^10 + ...
G.f. = q^9 - q^25 - 2*q^41 + 2*q^73 + 2*q^89 + q^121 - 2*q^137 - 2*q^153 + ...
MATHEMATICA
a[ n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]*QPochhammer[-q^1, q^8]* QPochhammer[-q^7, q^8]*QPochhammer[q^8], {q, 0, n}];
PROG
(PARI) {a(n) = local(m, j); if( n<0, 0, m = 16*n + 9; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==3 || (j%8==5)))))}
CROSSREFS
Cf. A030204.
Sequence in context: A307694 A106277 A088627 * A334841 A024713 A361166
KEYWORD
sign
AUTHOR
Michael Somos, Oct 11 2013
STATUS
approved