A226194 Expansion of f(-x^1, -x^7) * f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 0, -1, 1, -1, 1, -1, 0, 0, 2, 0, 1, -1, 1, -2, 0, 0, 1, -1, 0, -1, 1, 0, 1, -2, 0, -2, 1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 0, -1, 3, -1, 0, -1, 0, -2, 1, 0, 1, -1, 1, 0, 1, 0, 0, -2, 0, -1, 0, -1, 2, -2, 0, -1, 0, 0, 2, -1, 1, -1, 2, 0, 0, 0, 0, -1, 1, 0, 2

Offset

0,11

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/8) * eta(q) * eta(q^8)^2 / eta(q^2) in powers of q.

Euler transform of period 8 sequence [-1, 0, -1, 0, -1, 0, -1, -2, ...].

a(n) = -I/2 * b(8*n + 5) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (-1)^(e * (p+1)/8) * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = (e+1) * I^(e * (p-1)/4) if p == 1 (mod 4).

G.f.: Product_{k>0} (1 - x^(8*k))^2 / (1 + x^k).

a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = -a(n).

a(n) = (-1)^n * A053692(n).

Example

G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^10 + x^12 - x^13 + x^14 - 2*x^15 + ...
G.f. = q^5 - q^13 - q^29 + q^37 - q^45 + q^53 - q^61 + 2*q^85 + q^101 - q^109 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^8]^2 / QPochhammer[ -q, q], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A)^2 / eta(x^2 + A), n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 5; A = factor(n); simplify( -I/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4 == 3, if( e%2, 0, (-1)^(e * (p+1) / 8)), (e+1) * I^(e * (p-1) / 4)))))};

Crossrefs

Cf. A053692, A226192.

Sequence in context: A210638 A272903 A321458 * A053692 A341024 A286934

Adjacent sequences: A226191 A226192 A226193 * A226195 A226196 A226197

Keyword

sign

Author

Michael Somos, May 30 2013

Status

approved