A144377 Expansion of phi(q) / phi(q^5) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 2, -2, -4, 0, 0, -2, 4, 8, 0, 0, 4, -8, -14, 0, 0, -8, 14, 24, 0, 0, 12, -22, -40, 0, 0, -20, 36, 64, 0, 0, 32, -56, -98, 0, 0, -48, 84, 148, 0, 0, 72, -126, -220, 0, 0, -106, 184, 320, 0, 0, 152, -264, -460, 0, 0, -216, 376, 652, 0, 0, 306, -528

Offset

0,2

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. 235, Entry 67.

Links

Table of n, a(n) for n=0..65.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^5) in powers of q.

Euler transform of period 20 sequence [ 2, -3, 2, -1, 0, -3, 2, -1, 2, 0, 2, -1, 2, -3, 0, -1, 2, -3, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^4 - 2*u^2 +5) * (v^4 - 2*v^2 + 5) - 4 * (u^2 - 2*u*v - v^2)^2.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261968.

G.f.: Product_{k>0} P(20, x^k)^2 / (P(10, x^k)^3 * P(5, x^k)) where P(n, x) is the n-th cyclotomic polynomial.

a(5*n + 2) = a(5*n + 3) = 0.

a(n) = (-1)^n * A138527(n). Convolution inverse is A261968. - Michael Somos, Sep 06 2015

Example

G.f. = 1 + 2*q + 2*q^4 - 2*q^5 - 4*q^6 - 2*q^9 + 4*q^10 + 8*q^11 + 4*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5), n))};

Crossrefs

Cf. A138527, A261968.

Sequence in context: A273514 A048866 A262904 * A138527 A033718 A033737

Adjacent sequences: A144374 A144375 A144376 * A144378 A144379 A144380

Keyword

sign

Author

Michael Somos, Sep 18 2008

Status

approved