0,2

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. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 24 2002

Expansion of q^(-1/4)(1/2)k^(1/2) in powers of q.

Expansion of (1/q)(1/2)(1-sqrt(k'))/(1+sqrt(k')) in powers of q^4.

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

G.f. A(x) satisfies A(x)^2=A(x^2)/(1+4*x*A(x^2)^2). - Michael Somos Mar 19 2004

Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2(1+4v^2)-v . - Michael Somos Jul 09 2005

Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u3*(u6+u2)^2-u2*u6 . - Michael Somos Jul 09 2005

G.f.: (Product_{k>0} (1+x^(2k))/(1+x^(2k-1)))^2 = (Product_{k>0} (1-x^(4k))/(1-(-x)^k))^2.

Expansion of continued fraction 1/(1-x^2+(x^1+x^3)^2/(1-x^6+(x^2+x^6)^2/(1-x^10+(x^3+x^9)^2/...))). - Michael Somos Sep 01 2005

Given g.f. A(x), then B(x)=2*x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(1-u^4)(1-v^4)-(1-uv)^4 . - Michael Somos Jan 01 2006

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

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

q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...

(PARI) {a(n)=local(N, A); if(n<0, 0, N=(sqrtint(16*n+1)+1)\2; A=contfracpnqn( matrix(2, N, i, j, if(i==1, if(j<2, 1+O(x^(N^2+N)), (x^(j-1)+x^(3*j-3))^2), 1-x^(4*j-2)))); polcoeff(A[2, 1]/A[1, 1], 4*n))} /* Michael Somos Sep 01 2005 */

(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(A, 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)^2, n))}

Convolution inverse of A029839. Convolution square of A083365.

See A127391, A127392, A001936 for other versions of this sequence.

a(n)=(-1)^n A001936(n).

Cf. A002103, A046897.

Sequence in context: A034350 A006327 A103577 this_sequence A001936 A127297 A018739

Adjacent sequences: A079003 A079004 A079005 this_sequence A079007 A079008 A079009

sign,easy

Michael Somos, Dec 22 2002