A138532 Expansion of psi(x) / psi(x^5) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 2, 0, 0, 1, 0, -2, -1, 0, -2, 0, 3, 2, 0, 3, 0, -5, -2, 0, -3, 0, 6, 2, 0, 4, 0, -8, -3, 0, -6, 0, 11, 5, 0, 8, 0, -14, -6, 0, -10, 0, 18, 6, 0, 12, 0, -22, -9, 0, -16, 0, 28, 13, 0, 21, 0, -36, -14, 0, -25, 0, 44, 16, 0, 30, 0

Offset

0,11

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 233, Entry 66.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/2) * (eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q) in powers of q.

Euler transform of period 10 sequence [ 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - u^2)^2 - (u^2 - 1) * (u^2 - 5) * v^2.

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (u^2 - 1) * (v^2 - 5).

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * w * (v^2 - 1) - v * (v + w)^2.

Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u2 * u6 * (u3^2 - u1^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138526.

G.f.: (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-1) where P(n,x) is the n-th cyclotomic polynomial.

a(5*n + 2) = a(5*n + 4) = 0.

Convolution square is A138516. Convolution inverse is A116494.

Example

G.f. = 1 + x + x^3 - x^5 - x^8 + 2*x^10 + x^13 - 2*x^15 - x^16 - 2*x^18 + ...
G.f. = 1/q + q + q^5 - q^9 - q^15 + 2*q^19 + q^25 - 2*q^29 - q^31 - 2*q^35 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, 0, x^(1/2)] / EllipticTheta[ 2, 0, x^(5/2)], {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) / eta(x + A) * (eta(x^2 + A) / eta(x^10 + A))^2, n))};

Crossrefs

Cf. A116494, A138516.

Sequence in context: A084143 A025888 A145708 * A339445 A065293 A364377

Adjacent sequences: A138529 A138530 A138531 * A138533 A138534 A138535

Keyword

sign

Author

Michael Somos, Mar 23 2008

Status

approved