A131794 Expansion of q * psi(q) * psi(q^15) / (psi(q^3) * psi(q^5)) in powers of q where psi() is a Ramanujan theta function.

1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, 0, 1, 2, 1, -2, -3, -1, 1, 2, 3, 0, -3, -1, 2, 2, 0, -2, -6, -3, 4, 7, 3, -2, -5, -6, 2, 8, 3, -5, -6, -2, 4, 12, 7, -10, -15, -6, 5, 13, 12, -4, -18, -7, 11, 14, 6, -10, -24, -14, 20, 32, 12, -12, -29, -24, 9, 36, 15, -22, -30, -13, 22, 50, 27, -36, -63, -26, 24, 56, 45, -22, -69, -30, 42, 62, 27

Offset

1,14

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^5) * eta(q^30^2) / (eta(q) * eta(q^6)^2 * eta(q^10)^2 * eta(q^15)) in powers of q.

Euler transform of period 30 sequence [ 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 1, -1, 0, -1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, ...].

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

G.f.: x * Product_{k>0} P_15(x^k) * P_30(x^k)^2 where P_n() is the n-th cyclotomic polynomial.

a(n) = A131796(n) = -A131797(n) unless n = 0.

Example

G.f. = q + q^2 - q^5 - q^6 + q^8 - q^10 + q^13 + 2*q^14 + q^15 - 2*q^16 - 3*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]EllipticTheta[ 2, 0, q^(15/2)] / (EllipticTheta[ 2, 0, q^(3/2)] EllipticTheta[ 2, 0, q^(5/2)]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^15 + A)) * (eta(x^2 + A) * eta(x^30 + A) / ( eta(x^6 + A) * eta(x^10 + A)))^2, n))};

Crossrefs

Cf. A131796, A131797.

Sequence in context: A131797 A145727 A131796 * A145726 A322984 A277822

Adjacent sequences: A131791 A131792 A131793 * A131795 A131796 A131797

Keyword

sign

Author

Michael Somos, Jul 16 2007

Status

approved