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A232635 Expansion of psi(x) * phi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A232635 #11 Feb 16 2025 08:33:20
%S A232635 1,1,4,5,4,8,1,4,8,4,13,12,4,8,8,1,12,8,8,12,16,13,4,16,4,12,20,8,5,
%T A232635 12,12,12,16,8,8,20,17,16,16,8,20,24,8,8,16,1,24,16,8,12,20,16,12,28,
%U A232635 12,33,20,8,16,12,16,20,24,8,16,32,1,12,32,8,16,32,8
%N A232635 Expansion of psi(x) * phi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A232635 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A232635 Theta function of cubic lattice with respect to point (1/4, 0, 0).
%H A232635 G. C. Greubel, <a href="/A232635/b232635.txt">Table of n, a(n) for n = 0..1000</a>
%H A232635 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A232635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A232635 Expansion of q^(-1/8) * eta(q^4)^10 / (eta(q) * eta(q^2)^2 * eta(q^8)^4) in powers of q.
%F A232635 Euler transform of period 8 sequence [ 1, 3, 1, -7, 1, 3, 1, -3, ...].
%F A232635 a(3*n + 2) = 4 * A213023(n).
%e A232635 G.f. = 1 + x + 4*x^2 + 5*x^3 + 4*x^4 + 8*x^5 + x^6 + 4*x^7 + 8*x^8 + 4*x^9 + ...
%e A232635 G.f. = q + q^9 + 4*q^17 + 5*q^25 + 4*q^33 + 8*q^41 + q^49 + 4*q^57 + 8*q^65 + ...
%t A232635 a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^10 / (QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^8]^4), {q, 0, n}]
%o A232635 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^10 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^4), n))}
%Y A232635 Cf. A213023.
%K A232635 nonn
%O A232635 0,3
%A A232635 _Michael Somos_, Nov 27 2013