0,2

T. Miwa, Integrable Lattice Models and Branching Coefficients, Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), 862-870, Amer. Math. Soc., Providence, RI, 1987. MR0934288 (89h:82051)

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=-2*u^2+v^4+u^4*v^4.

G.f. A(x) satisfies 0=f(A(x), A(x^3)) where f(u, v)=u^4+2*u*v-2*u^3*v^3-v^4.

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-2*u+v*w^2+u^2*v*w^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*u2*u6 - 2*u1*u3 + u2*u3^2*u6.

Expansion of eta(q)^2 eta(q^4)/eta(q^2)^3 in powers of q.

Expansion of (1-k^2)^(1/8) = k'^(1/4) in powers of q=exp(-pi K'/K).

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

Expansion of (theta_4(q)/theta_3(q))^(1/2) = (phi(-q)/phi(q))^(1/2) = chi(-q)/chi(q) = psi(-q)/psi(q) = f(-q)/f(q) where phi, chi, psi, f are Ramanujan's theta functions.

G.f. A(x) satisfies 0=f(A(x), A(x^7)) where f(u, v)=(1-u^8)(1-v^8)-(1-uv)^8 . - Michael Somos Jan 01 2006

G.f. A(x) satisfies 0=f(A(x), A(x^5)) where f(u, v)=4*u*v*(1-u^4)*(1+v^4) -(v^2-u^2)*(u+v)^4 . - Michael Somos Sep 11 2006

(PARI) {a(n)=if(n<0, 0, polcoeff( prod(k=1, (n+1)\2, (1-x^(2*k-1))/(1+x^(2*k-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)^2*eta(x^4+A)/eta(x^2+A)^3, n))}

a(n)=(-1)^n A080054(n).

Adjacent sequences: A108491 A108492 A108493 this_sequence A108495 A108496 A108497

Sequence in context: A051466 A080015 A080054 this_sequence A078578 A018129 A091915

sign

Michael Somos, Jun 06 2005