A138526 Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

1, 2, 4, 8, 14, 22, 36, 56, 84, 126, 184, 264, 376, 528, 732, 1008, 1374, 1856, 2492, 3320, 4394, 5784, 7568, 9848, 12756, 16442, 21096, 26960, 34312, 43500, 54956, 69184, 86804, 108576, 135392, 168336, 208722, 258096, 318320, 391632, 480664

Offset

0,2

Comments

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

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

Euler transform of period 10 sequence [ 2, 1, 2, 1, 0, 1, 2, 1, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v^2)^2 - u^2 * (v^2 - 1) * (5*v^2 - 1).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - v^2) * (u + v)^2 - u * v * (u^2 - 1) * (5*v^2 - 1).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 * w^2 - u * v * (v^2 - 1).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u1 * u3 * (u2^2 - u6^2).

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 A138532.

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

Convolution square is A138517. Convolution inverse is A138527.

a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(3/4)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015

Example

G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 22*q^5 + 36*q^6 + 56*q^7 + 84*q^8 + ...

Mathematica

nmax=50; CoefficientList[Series[Product[(1+x^k)*(1-x^(5*k))/((1-x^k)*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)

Prog

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

Crossrefs

Cf. A138517, A138527, A138532.

Sequence in context: A231429 A259392 A261968 * A286522 A201347 A089054

Adjacent sequences: A138523 A138524 A138525 * A138527 A138528 A138529

Keyword

nonn

Author

Michael Somos, Mar 23 2008

Status

approved