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%I #46 Sep 08 2022 08:45:17
%S 1,-4,4,0,4,-8,0,0,4,-4,8,0,0,-8,0,0,4,-8,4,0,8,0,0,0,0,-12,8,0,0,-8,
%T 0,0,4,0,8,0,4,-8,0,0,8,-8,0,0,0,-8,0,0,0,-4,12,0,8,-8,0,0,0,0,8,0,0,
%U -8,0,0,4,-16,0,0,8,0,0,0,4,-8,8,0,0,0,0,0,8
%N Expansion of theta_4(q)^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
%C In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
%H Seiichi Manyama, <a href="/A104794/b104794.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.
%F Expansion of K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - _Michael Somos_, Jun 08 2015
%F Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
%F Euler transform of period 2 sequence [ -4, -2, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
%F Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.
%F G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
%F G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - _Michael Somos_, Jun 08 2015
%F a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n)). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).
%F a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - _Michael Somos_, Jun 08 2015
%F Convolution inverse of A001934. Convolution with A000729 is A227695. - _Michael Somos_, Jun 08 2015
%F G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - _Michael Somos_, Nov 05 2015
%F a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 02 2017
%F G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - _Ilya Gutkovskiy_, Sep 19 2018
%e G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* _Michael Somos_, Jun 06 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, Jun 06 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 / eta(x^2 + A)^2, n ))};
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* _Michael Somos_, Jan 31 2015 */
%o (Julia) # JacobiTheta4 is defined in A002448.
%o A104794List(len) = JacobiTheta4(len, 2)
%o A104794List(102) |> println # _Peter Luschny_, Mar 12 2018
%Y Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.
%K sign
%O 0,2
%A _Michael Somos_, Mar 26 2005
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