A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

1, -1, 1, 0, 1, -2, 0, 0, 1, -4, 2, 0, 0, -2, 0, 0, 1, -2, 4, 0, 2, 0, 0, 0, 0, -3, 2, 0, 0, -2, 0, 0, 1, 0, 2, 0, 4, -2, 0, 0, 2, -2, 0, 0, 0, -8, 0, 0, 0, -1, 3, 0, 2, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 1, -4, 0, 0, 2, 0, 0, 0, 4, -2, 2, 0, 0, 0, 0, 0, 2

Offset

0,6

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q) * eta(q^6)^6 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)^2) in powers of q.

Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 0, 1, -1, -3, -1, 1, 1, 1, -1, -2, ...].

Moebius transform is a period 72 sequence.

a(n) = (-1)^n * A258256(n). a(2*n) = A258256(n). a(3*n + 1) = - A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0.

a(n) = -b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -1 if e>0, b(3^e) = 2 * (1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015

Example

G.f. = 1 - q + q^2 + q^4 - 2*q^5 + q^8 - 4*q^9 + 2*q^10 - 2*q^13 + q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3 / (2 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[#, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + #) &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[ n==0 ], -Times @@ (Which[ # == 1, 1, # == 2, -1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 06 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 * eta(x^9 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)^2), n))};

Crossrefs

Cf. A258256, A258277, A258278.

Sequence in context: A301344 A301579 A072458 * A258256 A361290 A285638

Adjacent sequences: A256279 A256280 A256281 * A256283 A256284 A256285

Keyword

sign

Author

Michael Somos, Jun 02 2015

Status

approved