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A045828 One fourth of theta series of cubic lattice with respect to face.

%I A045828 #28 Feb 16 2025 08:32:38
%S A045828 1,2,2,4,3,2,6,4,4,6,4,4,7,8,2,8,8,4,10,4,4,10,10,8,9,4,6,12,8,6,10,
%T A045828 12,4,14,8,4,16,10,8,8,9,10,12,12,8,12,12,4,20,10,6,20,8,6,10,12,8,20,
%U A045828 18,8,11,12,12,16,8,6,20,16,12,14,8,12,20,14,6,12,20,8,26,12,8,22,8,12,15
%N A045828 One fourth of theta series of cubic lattice with respect to face.
%C A045828 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A045828 Number of solutions to n = t1 + t2 + 2*t3 where  t1, t2, t3 are triangular numbers. - _Michael Somos_, Jan 02 2006
%C A045828 The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - _Michael Somos_, Jun 29 2012
%D A045828 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
%H A045828 Seiichi Manyama, <a href="/A045828/b045828.txt">Table of n, a(n) for n = 0..10000</a>
%H A045828 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A045828 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A045828 Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - _Michael Somos_, Jan 02 2006
%F A045828 Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jun 29 2012
%F A045828 Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - _Michael Somos_, Mar 05 2003
%F A045828 Convolution of A033761 and A010054. - _Michael Somos_, Jun 29 2012
%F A045828 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - _Michael Somos_, Sep 08 2018
%e A045828 G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 2*x^5 + 6*x^6 + 4*x^7 + 4*x^8 + 6*x^9 + ...
%e A045828 G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + 3*q^9 + 2*q^11 + 6*q^13 + 4*q^15 + 4*q^17 + ...
%t A045828 a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* _Michael Somos_, Jun 29 2012 *)
%t A045828 a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* _Michael Somos_, Jun 29 2012 *)
%t A045828 QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%o A045828 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^2, n))}; /* _Michael Somos_, Oct 25 2006 */
%Y A045828 Cf. A005877, A005884, A033761, A010054, A033763, A212885.
%K A045828 nonn
%O A045828 0,2
%A A045828 _N. J. A. Sloane_
%E A045828 Edited by _Michael Somos_, Mar 05 2003