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A002448 Expansion of Jacobi theta function theta_4(x). 16
1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0 (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).

Number 2 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

REFERENCES

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.11), p. 6, Eq. (7.324).

J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.

J. W. L. Glaisher, On the deduction of series from infinite products, Messenger of Math., 2 (1873), p. 141.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Eric Weisstein's World of Mathematics, q-Series Identities

D. Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008

FORMULA

Expansion of phi(-q) in powers of q where phi() is a Ramanujan theta function.

Expansion of eta(q)^2 / eta(q^2) in powers of q. - Michael Somos, May 01 2003

Expansion of 2 * sqrt( k' * K / (2 pi) ) in powers of q. - Michael Somos, Nov 30 2013

Euler transform of period 2 sequence [ -2, -1, ...]. - Michael Somos, May 01 2003

G.f.: Sum_{k in Z} (-1)^k * x^(k^2) = Product_{k>0} (1 - x^k) / (1 + x^k). - Michael Somos, May 01 2003.

G.f.: 1 - 2 Sum_{k>0} x^k/(1 - x^k) Product_{j=1..k} (1 - x^j) / (1 + x^j). - Michael Somos, Apr 12 2012

a(n) = -2 * b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos, Jul 07 2006

a(3*n + 1) = -2 * A089802(n), a(9*n) = a(n). - Michael Somos, Jul 07 2006

a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A000122(n). a(n) = (-1)^n * A000122(n). a(8*n + 1) = -2 * A010054(n). - Michael Somos, Apr 12 2012

For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1)))*(-1)^(floor(sqrt(n)). - Mikael Aaltonen, Jan 17 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A010054. - Michael Somos, May 05 2016

EXAMPLE

G.f. = 1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 - 2*q^49 + ...

MAPLE

Sum((-x)^(m^2), m=-10..10);

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

QP = QPochhammer; s = QP[q]^2/QP[q^2] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

PROG

(PARI) {a(n) = if( n<0, 0, (-1)^n * issquare(n) * 2 - (n==0))}; /* Michael Somos, Jun 17 1999 */

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A), n))}; /* Michael Somos, May 01 2003 */

CROSSREFS

Cf. A000122, A010054, A089802.

Sequence in context: A139380 A128771 A000122 * A033759 A033755 A033753

Adjacent sequences:  A002445 A002446 A002447 * A002449 A002450 A002451

KEYWORD

sign,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 30 20:11 EDT 2016. Contains 275970 sequences.