A080332 G.f.: Product_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_{n in Z} (6*n + 1) * x^(n*(3*n + 1)/2).

1, -5, 7, 0, 0, -11, 0, 13, 0, 0, 0, 0, -17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 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

J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 306.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.6); p. 84, Eq. (32.63).

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

G.f.: theta_4(x)^2 * (Sum_{n in Z} (-1)^n * x^(n*(3*n + 1)/2)).

Expansion of f(-x)^2 * phi(x) = f(-x^2) * phi(-x^2)^2 in powers of x^2 where phi(), f() are Ramanujan theta functions. - Michael Somos, Feb 18 2003

Expansion of q^(-1/24) * eta(q)^5 / eta(q^2)^2 in powers of q.

Euler transform of period 2 sequence [-5, -3, ...]. - Michael Somos, Sep 09 2007

a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2* p^(e/2) if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 5 (mod 6). - Michael Somos, May 26 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113277. - Michael Somos, Feb 18 2003

a(5*n + 3) = a(5*n + 4) = a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0. a(25*n + 1) = -5 * a(n). - Michael Somos, Feb 18 2003

Example

G.f. = 1 - 5*x + 7*x^2 - 11*x^5 + 13*x^7 - 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 + 7*q^49 - 11*q^121 + 13*q^169 - 17*q^289 + 19*q^361 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^2]^2, {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 3, 0, x], {x, 0, 2 n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ -3, m], 0]]; (* Michael Somos, Mar 11 2015 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, if( p%6 == 1, p, -p)^(e\2))))}; /* Michael Somos, May 26 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^2 + A)^2, n))};
(PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -3, n), 0)};

Crossrefs

Cf. A010815, A113277.

Sequence in context: A355022 A133079 A116916 * A134756 A178902 A176713

Adjacent sequences: A080329 A080330 A080331 * A080333 A080334 A080335

Keyword

sign,easy

Author

Michael Somos, Feb 18 2003

Extensions

Definition changed by N. J. A. Sloane, Aug 14 2007

Status

approved