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A258196
Expansion of f(-x^2) * phi(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.
2
1, 0, -1, -2, -1, 2, 0, 2, 0, 0, 1, 0, 2, -2, -1, 0, -2, -2, 0, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 2, -1, 2, 0, 2, 0, 0, -2, -2, 0, 0, 0, -2, -2, 0, 1, 0, 0, -2, 2, 0, -2, 2, -1, 0, 0, -2, 2, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, -1, -2, -2, 2, 0, -2, 0, 2, -2
OFFSET
0,4
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
Expansion of q^(-1/12) * eta(q^2) * eta(q^3)^2 / eta(q^6) in powers of q.
Euler transform of period 6 sequence [ 0, -1, -2, -1, 0, -2, ...].
G.f.: Product_{k>0} (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k)).
a(49*n + 18) = a(49*n + 25) = a(49*n + 32) = a(49*n + 39) = a(49*n + 46) = 0.
EXAMPLE
G.f. = 1 - x^2 - 2*x^3 - x^4 + 2*x^5 + 2*x^7 + x^10 + 2*x^12 - 2*x^13 + ...
G.f. = q - q^25 - 2*q^37 - q^49 + 2*q^61 + 2*q^85 + q^121 + 2*q^145 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] EllipticTheta[ 4, 0, x^3], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / eta(x^6 + A), n))};
(PARI) {a(n) = my(A, p, e, x, w=quadgen(-8)); if( n<0, 0, n = 12*n + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%12 == 11, !(e%2), p%12 == 5, forstep(y = 3, sqrtint(2*p), 6, if( issquare(2*p - y^2, &x), if( x%6==5, x=-x); x = (x-1)/6; break)); (-1)^(e*x) * [1, w, -1, 0, -1, -w, 1, 0][e%8+1], x=0; forstep(y = 0, sqrtint(p), 6, if( issquare(p - y^2, &x), if( x%6==5, x=-x); x = [x-1, y]/6; break)); if( x==0, (-1)^(e\2) * !(e%2), (-1)^(e*(x[1] + x[2])) * (e+1))))))};
(PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^3)^2/eta(q^6)) \\ Altug Alkan, Aug 02 2018
CROSSREFS
Cf. A204770.
Sequence in context: A245472 A375158 A332685 * A159955 A327306 A053838
KEYWORD
sign
AUTHOR
Michael Somos, May 23 2015
STATUS
approved