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 A045834 Half of theta series of cubic lattice with respect to edge. 15
 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, 16, 20, 13, 24, 24, 8, 16, 16, 16, 28, 24, 12, 32, 16, 13, 28, 8, 20, 32, 32, 8, 20, 24, 16, 40, 16, 16, 32, 25, 20, 24, 24, 24, 28, 24, 8, 32, 36, 16, 44, 16, 12, 40, 32, 17, 36, 32 (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). 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. 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 Cf. A005876. Sequence in context: A232635 A201296 A246954 * A106148 A192038 A046577 Adjacent sequences:  A045831 A045832 A045833 * A045835 A045836 A045837 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 13 08:18 EST 2019. Contains 329093 sequences. (Running on oeis4.)