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A106508 Expansion of psi(x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

%I A106508 #18 Feb 16 2025 08:32:57
%S A106508 1,4,4,0,2,0,-8,0,-5,-16,4,0,-10,0,-8,0,9,8,0,0,14,0,16,0,-10,32,4,0,
%T A106508 0,0,8,0,14,-20,-20,0,2,0,0,0,-11,-16,-20,0,-32,0,16,0,0,-40,4,0,14,0,
%U A106508 -8,0,-9,32,-20,0,26,0,0,0,2,36,28,0,0,0,16,0,16,0,28,0,-22,0,0,0,14,56,-16,0,0,0,-40,0,0
%N A106508 Expansion of psi(x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
%C A106508 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A106508 G. C. Greubel, <a href="/A106508/b106508.txt">Table of n, a(n) for n = 0..1000</a>
%H A106508 C. Adiga, N. Anitha and T. Kim, <a href="https://arxiv.org/abs/math/0501528">Transformations of Ramanujan's Summation Formula and its Applications</a>, arXiv:math/0501528 [math.NT], 2005. See page 5.
%H A106508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A106508 Expansion of q^(-1/3) * eta(q^2)^10 / (eta(q)^4 * eta(q^4)^2) in powers of q.
%F A106508 Euler transform of period 4 sequence [4, -6, 4, -4, ...].
%F A106508 a(n) = (-1)^n * A187149(n). a(4*n + 3) = a(8*n + 5) = 0.
%F A106508 G.f. Product_{k>0} (1 + x^k)^4 (1 - x^(2*k))^4 / (1 + x^(2*k))^2.
%e A106508 1 + 4*x + 4*x^2 + 2*x^4 - 8*x^6 - 5*x^8 - 16*x^9 + 4*x^10 - 10*x^12 + ...
%e A106508 q + 4*q^4 + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 - 16*q^28 + 4*q^31 - 10*q^37 + ...
%t A106508 a[n_]:= SeriesCoefficient[(x^(-1/2)/16)*EllipticTheta[2, 0, x^(1/2)]^4* QPochhammer[x^2, x^4]^2, {x, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o A106508 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / (eta(x^4 + A)^2 * eta(x + A)^4), n))}
%Y A106508 Cf. A187149.
%K A106508 sign
%O A106508 0,2
%A A106508 _Michael Somos_, May 24 2005