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A213622 Expansion of phi(x) * psi(x) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A213622 #12 Feb 16 2025 08:33:17
%S A213622 1,3,4,7,8,4,9,8,4,16,9,8,20,8,8,11,8,12,20,20,8,15,16,12,20,16,8,24,
%T A213622 21,8,20,8,16,28,24,8,17,32,12,36,16,8,24,16,24,19,20,20,32,16,12,28,
%U A213622 16,20,44,27,12,36,24,16,28,24,16,28,32,12,25,32,12,48
%N A213622 Expansion of phi(x) * psi(x) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A213622 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213622 G. C. Greubel, <a href="/A213622/b213622.txt">Table of n, a(n) for n = 0..1000</a>
%H A213622 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213622 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213622 Expansion of q^(-1/8) * eta(q^2)^5 * eta(q^4)^3 / (eta(q)^3 * eta(q^8)^2), in powers of q.
%F A213622 Euler transform of period 8 sequence [ 3, -2, 3, -5, 3, -2, 3, -3, ...].
%e A213622 1 + 3*x + 4*x^2 + 7*x^3 + 8*x^4 + 4*x^5 + 9*x^6 + 8*x^7 + 4*x^8 + ...
%e A213622 q + 3*q^9 + 4*q^17 + 7*q^25 + 8*q^33 + 4*q^41 + 9*q^49 + 8*q^57 + 4*q^65 + ...
%t A213622 a[ n_] := SeriesCoefficient[ 1/2 EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, 2 n + 1/4}]; Table[a[n], {n, 0, 80}]
%o A213622 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^4 + A)^3 / (eta(x + A)^3 * eta(x^8 + A)^2), n))}
%K A213622 nonn
%O A213622 0,2
%A A213622 _Michael Somos_, Jun 16 2012