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A045836 Half of theta series of b.c.c. lattice with respect to long edge.

%I A045836 #26 Feb 16 2025 08:32:38
%S A045836 1,2,0,0,4,4,0,0,5,4,0,0,4,8,0,0,8,6,0,0,8,4,0,0,5,12,0,0,12,8,0,0,8,
%T A045836 8,0,0,4,12,0,0,16,8,0,0,12,8,0,0,9,14,0,0,12,16,0,0,8,4,0,0,12,16,0,
%U A045836 0,16,16,0,0,16,8,0,0,8,20,0,0,16,8,0,0,17
%N A045836 Half of theta series of b.c.c. lattice with respect to long edge.
%C A045836 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A045836 The body-centered cubic (b.c.c. also known as D3*) lattice is the set of all triples [a, b, c] where the entries are all integers or all one half an odd integer. A long edge is centered at a triple with two integer entries and the remaining entry is one half an odd integer. - _Michael Somos_, May 31 2012
%H A045836 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A045836 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A045836 From _Michael Somos_, May 31 2012: (Start)
%F A045836 Expansion of x * phi(x) * psi(x^4)^2 = x * psi(-x^2)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
%F A045836 Expansion of eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.
%F A045836 Euler transform of period 8 sequence [ 2, -3, 2, 1, 2, -3, 2, -3, ...].
%F A045836 a(4*n) = a(4*n + 3) = 0. a(n) = A004025(n) / 2. a(4*n + 1) = A045834(n). a(4*n + 2) = 2 * A045828(n). (End)
%e A045836 q + 2*q^2 + 4*q^5 + 4*q^6 + 5*q^9 + 4*q^10 + 4*q^13 + 8*q^14 + 8*q^17 + ...
%t A045836 a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[QPochhammer[x^2+A]^5 * (QPochhammer[x^8+A]^4 / (QPochhammer[x+A]^2*QPochhammer[x^4+A]^4)), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 05 2015, adapted from PARI *)
%o A045836 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^4 / (eta(x + A)^2 * eta(x^4 + A)^4), n))} /* _Michael Somos_, May 31 2012 */
%Y A045836 Cf. A004025, A045828, A045834.
%K A045836 nonn
%O A045836 1,2
%A A045836 _N. J. A. Sloane_