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A213022 Expansion of phi(x)^2 * psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A213022 #16 Feb 16 2025 08:33:17
%S A213022 1,5,8,5,8,16,9,8,16,8,17,24,8,16,16,13,24,16,16,24,32,13,8,32,8,24,
%T A213022 40,16,25,24,24,24,32,16,16,40,17,32,32,16,40,48,16,16,32,21,48,32,16,
%U A213022 24,40,32,24,56,24,45,40,16,32,24,32,40,48,16,32,64,25,24
%N A213022 Expansion of phi(x)^2 * psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A213022 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A213022 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.
%H A213022 G. C. Greubel, <a href="/A213022/b213022.txt">Table of n, a(n) for n = 0..1000</a>
%H A213022 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213022 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213022 Expansion of q^(-1/8) * eta(q^2)^12 / (eta(q)^5 * eta(q^4)^4) in powers of q.
%F A213022 Expansion of q^(-1/16) times theta series of b.c.c. lattice with respect to point [0, 0, 1/4] in powers of q^(1/2).
%F A213022 Euler transform of period 4 sequence [ 5, -7, 5, -3, ...].
%F A213022 6 * a(n) = A005875(8*n + 1).
%e A213022 a(0) = 1 since the norm squared of point [0, 0, 0] with respect to [0, 0, 1/4] is 1/16 = 1/16 + 1/2*0.
%e A213022 a(1) = 5 since the norm squared of points [-1/2, -1/2, -1/2], [-1/2, 1/2, -1/2], [0, 0, -1], [1/2, -1/2, -1/2], [1/2, 1/2, -1/2] with respect to [0, 0, 1/4] is 9/16 = 1/16 + 1/2*1.
%e A213022 1 + 5*x + 8*x^2 + 5*x^3 + 8*x^4 + 16*x^5 + 9*x^6 + 8*x^7 + 16*x^8 + 8*x^9 + ...
%e A213022 q + 5*q^9 + 8*q^17 + 5*q^25 + 8*q^33 + 16*q^41 + 9*q^49 + 8*q^57 + 16*q^65 + ...
%t A213022 CoefficientList[QPochhammer[q^2]^12/(QPochhammer[q]^5*QPochhammer[q^4]^4) + O[q]^70, q] (* _Jean-François Alcover_, Nov 05 2015 *)
%o A213022 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4), n))}
%Y A213022 Cf. A005875.
%K A213022 nonn
%O A213022 0,2
%A A213022 _Michael Somos_, Jun 03 2012