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A014705 Expansion of ((theta_2)^4 + (theta_3)^4) / eta(z/2)^4.

%I A014705 #23 Jun 01 2025 18:04:53
%S A014705 1,28,134,568,1809,5316,13990,34696,80724,180068,384940,796760,
%T A014705 1598789,3127360,5971922,11170160,20491033,36947444,65553412,
%U A014705 114619248,197681341,336670120,566630192,943234040,1553941445,2535325644,4098671374,6568931200,10441889389
%N A014705 Expansion of ((theta_2)^4 + (theta_3)^4) / eta(z/2)^4.
%C A014705 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A014705 G. C. Greubel, <a href="/A014705/b014705.txt">Table of n, a(n) for n = 0..1000</a>
%H A014705 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A014705 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A014705 Expansion of (phi(x)^4 + 16 * x* psi(x^2)^4) / f(-x)^4 in powers of x where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 30 2013
%F A014705 Expansion of (phi(x)^4 + phi(-x)^4) / (2 * f(-x^2)^4) = (chi(x)^8 + chi(-x)^8) / 2 in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 30 2013
%F A014705 a(n) = A101127(2*n). - _Michael Somos_, Sep 30 2013
%F A014705 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 19 2018
%e A014705 G.f. = 1 + 28*x + 134*x^2 + 568*x^3 + 1809*x^4 + 5316*x^5 + 13990*x^6 + ...
%e A014705 G.f. = 1/q + 28*q^5 + 134*q^11 + 568*q^17 + 1809*q^23 + 5316*q^29 + ...
%t A014705 a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2]^8 + QPochhammer[ -x, x^2]^8 ) / 2, {x, 0, 2 n}]; (* _Michael Somos_, Sep 30 2013 *)
%t A014705 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) / QPochhammer[ q]^4, {q, 0, n}]; (* _Michael Somos_, Sep 30 2013 *)
%o A014705 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^(2*k-1), 1 + x * O(x^(2*n)))^8, 2*n))}; /* _Michael Somos_, Sep 30 2013 */
%Y A014705 Cf. A101127.
%K A014705 nonn
%O A014705 0,2
%A A014705 _N. J. A. Sloane_