0,3

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eqs. (9.1),(9.3).

Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2)) where f(u, v)=v^2-u^4(1+4v^4).

Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^3)) where f(u, v)=v^4-u^4+uv+4(uv)^3.

Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=w-u^2v(1+2w^2) . Michael Somos May 29 2005

Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u2*u6-u1*u3*(u2^2+u6^2) . - Michael Somos May 29 2005

Given g.f. A(x), then B(x)=sqrt(2)*x*A(x^8) satisfies 0=f(B(x), B(x^7)) where f(u, v)=(1-u^8)(1-v^8)-(1-uv)^8 . - Michael Somos Jan 01 2006

G.f.: Product_{n>0} (1+x^(2n))/(1+x^(2n-1)) = (Sum_{k>0} x^(k^2-k))/(Sum_{k>0} x^((k^2-k)/2)).

Expansion of f(q) / f(-q^4) = phi(q) / psi(q) = psi(q) / psi(q^2) = phi(-q^2) / psi(-q) = chi(q) * chi(-q^2) = chi^2(q) * chi(-q) = chi^2(-q^2) / chi(-q) = (phi(q) / psi(q^2))^(1/2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is elliptic modulus and q is the nome.

Euler transform of period 4 sequence [ -1, 2, -1, 0, ...].

G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).

q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 + ...

(PARI) a(n)=local(A, m); if(n<0, 0, A=1+O(x); m=1; while(m<=n, m*=2; A=subst(A, x, x^2); A=sqrt(A/(1+4*x*A^2))); polcoeff(sqrt(A), n))

(PARI) {a(n)=local(A); if(n<0, 0, A=contfracpnqn(matrix(2, (sqrtint(8*n+1)+1)\2, i, j, if(i==1, x^(j-1), 1+if(j>1, x^(j-1))))); polcoeff(A[2, 1]/A[1, 1]+x*O(x^n), n))}

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3, n))}

A001935(n)=(-1)^n a(n). Convolution square is A079006.

Sequence in context: A073576 A069907 A001935 this_sequence A007604 A013950 A018550

Adjacent sequences: A083362 A083363 A083364 this_sequence A083366 A083367 A083368

sign

Michael Somos, Apr 24 2003