A145726 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) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60) / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30)) in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

G.f.: x * Product_{k>0} P(15, x^k) * P(60, x^k) where P(n, x) is the n-th cyclotomic polynomial.

a(n) = A145727(n) unless n=0. a(n) = -(-1)^n * A131794(n). a(2*n) = - A094022(n). Convolution inverse of A145725.

Example

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

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A)), n))}

Crossrefs

Cf. A094022, A131794, A145725, A145727.

Sequence in context: A145727 A131796 A131794 * A322984 A277822 A327616

Adjacent sequences: A145723 A145724 A145725 * A145727 A145728 A145729

Keyword

sign

Author

Michael Somos, Oct 23 2008

Status

approved