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A263452 Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() are Ramanujan theta functions.

%I A263452 #12 Feb 16 2025 08:33:27
%S A263452 1,1,2,0,2,1,2,0,1,2,2,0,3,1,4,0,5,3,2,0,3,3,4,0,4,2,4,0,3,2,4,0,4,2,
%T A263452 4,0,5,5,4,0,3,3,8,0,7,3,6,0,4,4,4,0,6,4,4,0,9,3,6,0,4,4,4,0,4,3,8,0,
%U A263452 5,5,6,0,9,3,4,0,7,6,6,0,7,6,10,0,6,3
%N A263452 Expansion of f(-q^3)^3 * psi(q^12) / f(-q) in powers of q where ps(), f() are Ramanujan theta functions.
%C A263452 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263452 G. C. Greubel, <a href="/A263452/b263452.txt">Table of n, a(n) for n = 0..2500</a>
%H A263452 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263452 Expansion of q^(-11/6) * eta(q^3)^3 * eta(q^24)^2 / (eta(q) * eta(q^12)) in powers of q.
%F A263452 Euler transform of period 24 sequence [ 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -1, 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, 1, -3, ...].
%F A263452 2 * a(n) = A261444(2*n + 1).  a(4*n + 1) = A212907(n). a(4*n + 3) = 0.
%F A263452 -2 * a(n) = A263527(2*n + 3). - _Michael Somos_, Nov 05 2015
%e A263452 G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + ...
%e A263452 G.f. = q^11 + q^17 + 2*q^23 + 2*q^35 + q^41 + 2*q^47 + q^59 + 2*q^65 + ...
%t A263452 a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 EllipticTheta[ 2, 0, q^6] / ( 2 q^(3/2) QPochhammer[ q]), {q, 0, n}];
%o A263452 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^12 + A)), n))};
%Y A263452 Cf. A212907, A261444, A263527.
%K A263452 nonn
%O A263452 0,3
%A A263452 _Michael Somos_, Oct 18 2015