0,4

G.f.: A(x) = G(x^2) + x*G(x^2)^4 where G(x^2) = A(x)*A(-x) and G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

a(2n) = C(7*n,n)/(6*n+1); a(2n+1) = C(7*n+3,n)*4/(6*n+4).

G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 +...

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then

A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where

G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...

G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 +...

form the bisections of A(x).

By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where

A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 +...

A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+...

(PARI) {a(n)=local(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^4*subst(A^3, x, -x)); polcoeff(A, n)}

(PARI) {a(n)=local(m=n\2, p=3*(n%2)+1); binomial(7*m+p-1, m)*p/(6*m+p)}

Cf. A002296 (bisection).

Sequence in context: A149089 A004031 A153062 this_sequence A149090 A103059 A123809

Adjacent sequences: A143544 A143545 A143546 this_sequence A143548 A143549 A143550

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2008