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A225872 Expansion of k(q)^3 * k'(q)^2 * (K(q) / (Pi/2))^6 / 64 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

%I A225872 #22 Feb 16 2025 08:33:19
%S A225872 0,1,-4,2,8,-13,28,-26,-56,69,-48,134,80,-182,-84,-312,280,204,332,
%T A225872 142,-816,91,-196,780,-224,-526,-244,-1198,2216,767,508,-390,-400,
%U A225872 -1167,-1424,466,-2264,1391,1392,3796,-1480,-11,1768,-2274,1320,-1508,-1984,-8450
%N A225872 Expansion of k(q)^3 * k'(q)^2 * (K(q) / (Pi/2))^6 / 64 in powers of q where k(), k'(), K() are Jacobi elliptic functions.
%C A225872 In Glaisher (1907) this is denoted by beta'(m) = beta(m)/16 on page 56 while beta(m) (see A322032) is defined on page 38.
%C A225872 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A225872 J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62.
%H A225872 G. C. Greubel, <a href="/A225872/b225872.txt">Table of n, a(n) for n = 0..1000</a>
%H A225872 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A225872 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A225872 Expansion of x * (psi(x) * psi(-x)^2)^4 in powers of x where psi() is a Ramanujan theta function.
%F A225872 Expansion of x * (f(-x) * f(-x^4)^2)^4 in powers of x where f() is a Ramanujan theta function.
%F A225872 Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2)^4 in powers of q.
%F A225872 Euler transform of period 4 sequence [ -4, -4, -4, -12, ...].
%F A225872 G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 16 * (t/i)^6 * g(t) where q = exp(2*Pi*i*t) and g() is the g.f. for A225912.
%F A225872 G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.
%F A225872 |a(n)| = A002291(n).
%e A225872 x - 4*x^2 + 2*x^3 + 8*x^4 - 13*x^5 + 28*x^6 - 26*x^7 - 56*x^8 + 69*x^9 + ...
%e A225872 q^3 - 4*q^5 + 2*q^7 + 8*q^9 - 13*q^11 + 28*q^13 - 26*q^15 - 56*q^17 + ...
%t A225872 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}]
%t A225872 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, I q]^2 )^4 / -4096, {q, 0, 2 n + 1}]
%o A225872 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2)^4, n))}
%Y A225872 Cf. A002291, A225912, A322032.
%K A225872 sign
%O A225872 0,3
%A A225872 _Michael Somos_, May 18 2013