A133089 Expansion of f(x)^3 in powers of x where f() is a Ramanujan theta function.

1, 3, 0, -5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

Links

Seiichi Manyama, 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 q^(-1/8) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^3 in powers of q.

Euler transform of period 4 sequence [ 3, -6, 3, -3, ...].

a(n) = b(8*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).

G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 64 (t/i)^(3/2) f(t) where q = exp(2 Pi i t).

a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = 3 * a(n). a(25*n + 3) = -5 * a(n).

G.f.: Sum_{k>=0} (-1)^floor(k/2) * (2*k + 1) * x^(k*(k + 1))/2.

G.f.: ( Product_{k>0} (1 - x^k) * (1 + x^k)^2 / (1 + x^(2*k)) )^3.

a(n) = -(-1)^n * A010816(n). a(3*n) = A133079(n).

Example

G.f. = 1 + 3*x - 5*x^3 - 7*x^6 + 9*x^10 + 11*x^15 - 13*x^21 - 15*x^28 + ...
G.f. = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3, {x, 0, n}]; (* Michael Somos, Jun 19 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, if( issquare( 8*n+1, &n), (-1)^( (n-1) \ 4) * n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^3, n))};

Crossrefs

Cf. A010816, A133079.

Sequence in context: A143073 A154725 A010816 * A198954 A136599 A227498

Adjacent sequences: A133086 A133087 A133088 * A133090 A133091 A133092

Keyword

sign

Author

Michael Somos, Sep 09 2007

Status

approved