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A225853
Expansion of phi(x) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.
1
1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 7, 8, 0, 0, 13, 14, 0, 0, 19, 20, 0, 0, 29, 34, 0, 0, 43, 46, 0, 0, 62, 70, 0, 0, 90, 96, 0, 0, 126, 138, 0, 0, 174, 186, 0, 0, 239, 262, 0, 0, 325, 346, 0, 0, 435, 472, 0, 0, 580, 620, 0, 0, 769, 826, 0, 0, 1007, 1072, 0
OFFSET
0,2
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 chi(x)^2 * chi(-x^2) = chi(x)^3 * chi(-x) = chi(-x^2)^3 / chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A029552.
G.f.: Product_{k>0} (1 - x^(4*k-2))^3 / (1 - x^(2*k-1))^2 = (Sum_{k in Z} x^k^2) / (Product_{k>0} (1 - x^(4*k))).
a(n) = (-1)^n * A143161(n). a(4*n + 2) = a(4*n + 3) = 0.
EXAMPLE
1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 7*x^12 + 8*x^13 + 13*x^16 + ...
1/q + 2*q^5 + 3*q^23 + 2*q^29 + 4*q^47 + 6*q^53 + 7*q^71 + 8*q^77 + 13*q^95 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]/QPochhammer[q^4], {q, 0, n}];
a[n_]:= SeriesCoefficient[QPochhammer[q^2, q^4]^3/QPochhammer[q, q^2]^2, {q, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3), n))}
CROSSREFS
Sequence in context: A224777 A259827 A143161 * A342128 A330463 A142886
KEYWORD
nonn
AUTHOR
Michael Somos, May 17 2013
STATUS
approved