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A045834 Half of theta series of cubic lattice with respect to edge.

%I A045834 #29 Feb 16 2025 08:32:38
%S A045834 1,4,5,4,8,8,5,12,8,4,16,12,9,12,8,12,16,16,8,16,17,8,24,8,8,28,16,12,
%T A045834 16,20,13,24,24,8,16,16,16,28,24,12,32,16,13,28,8,20,32,32,8,20,24,16,
%U A045834 40,16,16,32,25,20,24,24,24,28,24,8,32,36,16,44,16,12,40,32,17,36,32
%N A045834 Half of theta series of cubic lattice with respect to edge.
%C A045834 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A045834 Robert Israel, <a href="/A045834/b045834.txt">Table of n, a(n) for n = 0..10000</a>
%H A045834 J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 107.
%H A045834 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A045834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A045834 Euler transform of period 4 sequence [ 4, -5, 4, -3,...]. - _Michael Somos_, Feb 28 2006
%F A045834 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
%F A045834 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
%F A045834 G.f.: Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^3 / (1 + x^(2*k))^2. - _Michael Somos_, Feb 28 2006
%F A045834 Expansion of phi(x)^2 * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Oct 25 2006
%F A045834 A005876(n) = 2*a(n).
%e A045834 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 + ...
%e A045834 G.f. = q + 4*q^5 + 5*q^9 + 4*q^13 + 8*q^17 + 8*q^21 + 5*q^25 + 12*q^29 + ...
%p A045834 S:= series((1/8)*JacobiTheta2(0, sqrt(q))^2*(JacobiTheta3(0, q^(1/4))+JacobiTheta4(0, q^(1/4)))/q^(1/4), q, 1001):
%p A045834 seq(coeff(S,q,j),j=0..1000); # _Robert Israel_, Nov 13 2016
%t A045834 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 *)
%t A045834 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 *)
%o A045834 (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 */
%Y A045834 Cf. A005876.
%K A045834 nonn
%O A045834 0,2
%A A045834 _N. J. A. Sloane_