0,2

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(ab)=theta_4(q^2)^2.

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.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Euler transform of period 2 sequence [ -4, -2, ...].

Expansion of eta(q)^4/eta(q^2)^2 in powers of q.

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)-2uw^2.

G.f.: theta_4(q)^2 = (Sum_{k} (-q)^(k^2))^2 = (Product_{k>0} (1-q^(2k))(1-q^(2k-1))^2)^2.

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.

Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].

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 g.f. for A008441.

a(4*n+3) = 0.

1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

(PARI) a(n)=if(n<1, n==0, (-1)^n*4*sumdiv(n, d, (d%4==1)-(d%4==3)))

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^4/eta(x^2+A)^2, n))}

(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))}

A004018(n) = (-1)^n * a(n) = a(2*n). -4 * A008441(n) = a(4*n+1). -4 * A113652(n) = a(n) unless n=0. 4 * A122865(n) = a(6*n+2). 4 * A122856(n) = a(6*n+4). -4 * A113407(n) = a(8*n+1). -8 * A053692(n) = a(8*n+5).

Sequence in context: A138518 A155836 A164613 this_sequence A004018 A028658 A028642

Adjacent sequences: A104791 A104792 A104793 this_sequence A104795 A104796 A104797

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Michael Somos, Mar 26 2005