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A129588 Expansion of q^-1 * theta_2(q)^4 in powers of q^2.

%I A129588 #21 Feb 16 2025 08:33:05
%S A129588 16,64,96,128,208,192,224,384,288,320,512,384,496,640,480,512,768,768,
%T A129588 608,896,672,704,1248,768,912,1152,864,1152,1280,960,992,1664,1344,
%U A129588 1088,1536,1152,1184,1984,1536,1280,1936,1344,1728,1920,1440
%N A129588 Expansion of q^-1 * theta_2(q)^4 in powers of q^2.
%C A129588 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A129588 K. Bobek, Einleitung in die Theorie der elliptischen Funktionen, Teubner Leipzig, 1884, p. 101.
%H A129588 G. C. Greubel, <a href="/A129588/b129588.txt">Table of n, a(n) for n = 0..1000</a>
%H A129588 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A129588 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A129588 G.f. Sum_{k>=0} a(k)*q^(2*k + 1) = theta2(q)^4 = theta3(q)^4 - theta4(q)^4.
%F A129588 Expansion of 16 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Jun 11 2007
%F A129588 Number of solutions of 2*n + 1 = (x^2 + y^2 + z^2 + w^2) / 4 in odd integers. - _Michael Somos_, Jun 11 2007
%F A129588 G.f.: 16 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - _Michael Somos_, Jun 11 2007
%F A129588 a(n) = 16*A008438(n) = A000118(n) - A096727(n).
%e A129588 G.f. = 16 + 64*x + 96*x^2 + 128*x^3 + 208*x^4 + 192*x^5 + 224*x^6 + ...
%e A129588 G.f. = 16*q + 64*q^3 + 96*x^5 + 128*q^7 + 208*q^9 + 192*q^11 + 224*q^13 + ...
%t A129588 a[n_]:= SeriesCoefficient[q^(-1/2)*EllipticTheta[2, 0, q^(1/2)]^4, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 15 2018 *)
%t A129588 CoefficientList[Series[x^(-1/2)*EllipticTheta[2, 0, x^(1/2)]^4, {x, 0, 50}], x] (* _Vaclav Kotesovec_, Apr 16 2018 *)
%o A129588 (PARI) {a(n) = if( n<0, 0, 16 * sigma(2*n + 1))}; /* _Michael Somos_, Jun 11 2007 */
%Y A129588 Cf. A000118, A000122, A002448, A008438, A096727.
%K A129588 nonn
%O A129588 0,1
%A A129588 _Ralf Stephan_, May 30 2007