OFFSET
0,2
COMMENTS
LINKS
Robert Israel, 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. 107.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 4 sequence [ 4, -5, 4, -3,...]. - Michael Somos, Feb 28 2006
Expansion of theta_2(q^2)^2 * (theta_3(q) + theta_4(q)) / (8*q) in powers of q^4. - Michael Somos, Feb 28 2006
Expansion of q^(-1/4) * eta(q^2)^9 / (eta(q)^4 * eta(q^4)^2) in powers of q. - Michael Somos, Feb 28 2006
G.f.: Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^3 / (1 + x^(2*k))^2. - Michael Somos, Feb 28 2006
Expansion of phi(x)^2 * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
A005876(n) = 2*a(n).
EXAMPLE
G.f. = 1 + 4*x + 5*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 5*x^6 + 12*x^7 + 8*x^8 + ...
G.f. = q + 4*q^5 + 5*q^9 + 4*q^13 + 8*q^17 + 8*q^21 + 5*q^25 + 12*q^29 + ...
MAPLE
S:= series((1/8)*JacobiTheta2(0, sqrt(q))^2*(JacobiTheta3(0, q^(1/4))+JacobiTheta4(0, q^(1/4)))/q^(1/4), q, 1001):
seq(coeff(S, q, j), j=0..1000); # Robert Israel, Nov 13 2016
MATHEMATICA
s = EllipticTheta[3, 0, q]^2*EllipticTheta[2, 0, q]/(2*q^(1/4)) + O[q]^75; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from 5th formula *)
QP = QPochhammer; s = QP[q^2]^9/(QP[q]^4*QP[q^4]^2) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 +A)^9 / (eta(x + A)^4 * eta(x^4 + A)^2), n))}; /* Michael Somos, Feb 28 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved