0,4

Expansion of 1/psi(q) in powers of q where psi() is a Ramanujan theta function.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41

C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, See page 5

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

Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^4(w^4 +4v^4) -v^6w^2.

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

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

Sum_{k>=0} a(k)x^(8k-1) = 1/(Sum_{k} x^((4k+1)^2)).

Expansion of q^(1/8)* eta(q)/eta(q^2)^2 in powers of q.

G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan] (Michael Somos, Jul 21 2008)

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t / i)^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A015128. - Michael Somos Nov 01 2008

1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 +...

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

Convolution inverse of A010054. (-1)^n * A006950(n) = a(n).

Sequence in context: A014670 A036034 A006950 this_sequence A052335 A160333 A136422

Adjacent sequences: A106504 A106505 A106506 this_sequence A106508 A106509 A106510

sign

Michael Somos, May 04 2005

Definition changed by N. J. A. Sloane (njas(AT)research.att.com), Aug 14 2007