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A258322 Expansion of phi(-q) * phi(-q^9) in powers of q where phi() is a Ramanujan theta function. 3
1, -2, 0, 0, 2, 0, 0, 0, 0, -4, 4, 0, 0, -4, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, -4, 0, 0, 4, 0, 0, 0, 0, -8, 0, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, -4, 4 (list; graph; refs; listen; history; text; internal format)
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..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, -4, -1, -2, -1, -2, -1, -2, -1, -2, -2, ...]. - Michael Somos, May 26 2015

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

a(n) = -2 * A257900(n) unless n = 0 or n == 2 (mod 3).

a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 0. a(9*n) = -4 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.

EXAMPLE

G.f. = 1 - 2*q + 2*q^4 - 4*q^9 + 4*q^10 - 4*q^13 + 2*q^16 + 4*q^18 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^9]^2 / (QPochhammer[ q^2] QPochhammer[ q^18]), {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A) * eta(x^18 + A)), n))};

(PARI) {a(n) = my(A, p, e); if( n<1, n==0, if( n%3 == 2, 0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p%4==3, if(p==3, 2, 1)*(1-e%2), e+1))))};

CROSSREFS

Cf. A113652, A257900, A258277.

Sequence in context: A317641 A107497 A000095 * A258034 A243828 A034949

Adjacent sequences:  A258319 A258320 A258321 * A258323 A258324 A258325

KEYWORD

sign

AUTHOR

Michael Somos, May 26 2015

STATUS

approved

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Last modified June 16 12:38 EDT 2019. Contains 324152 sequences. (Running on oeis4.)