A143342 G.f. satisfies: A(x) = 1 + x*A(x)^5/A(-x).

1, 1, 6, 40, 374, 3215, 34298, 326360, 3710278, 37289620, 440121880, 4577214736, 55375589594, 589530372890, 7258264793564, 78597770766160, 980423896907046, 10754940952651740, 135521929778850952, 1501817992511869280

Offset

0,3

Comments

More generally, if A(x) = 1 + x*A(x)^n/A(-x)

then A(x) - x*A(x)^n = 1 + x^2*[A(x)*A(-x)]^(n-1)

so that a bisection of A(x) equals a bisection of A(x)^n.

Links

Table of n, a(n) for n=0..19.

Formula

G.f. satisfies: A(x) - x*A(x)^5 = 1 + x^2*[A(x)*A(-x)]^4.

G.f. satisfies:

_ A(x) = Sum_{n>=1} x^n * A(x)^(4*n)/A(-x)^n;

_ A(x) = exp( Sum_{n>=1} x^n/n * A(x)^(4*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]

Example

A bisection of g.f. A(x) equals a bisection of A(x)^5:
A(x) = 1 + x + 6*x^2 + 40*x^3 + 374*x^4 + 3215*x^5 + 34298*x^6 + 326360*x^7 +...
A(x)^5 = 1 + 5*x + 40*x^2 + 330*x^3 + 3215*x^4 + 30756*x^5 + 326360*x^6 +...
so that A(x) - x*A(x)^5 = 1 + x^2*[A(x)*A(-x)]^4, where
[A(x)*A(-x)]^4 = 1 + 44*x^2 + 3542*x^4 + 358468*x^6 + 40846025*x^8 + +...
A(x)*A(-x) = 1 + 11*x^2 + 704*x^4 + 65054*x^6 + 7062088*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^4/A(-x) + x^2*A(x)^8/A(-x)^2 + x^3*A(x)^12/A(-x)^3 +...
log(A(x)) = x*A(x)^4/A(-x) + x^2/2*A(x)^8/A(-x)^2*x^2 + x^3/3*A(x)^12/A(-x)^3 +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*A^5/subst(A, x, -x)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*A^(4*m)/subst(A^m, x, -x+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, A^(4*m)*subst(A^-m, x, -x)*x^m/m)+x*O(x^n))); polcoeff(A, n)}

Crossrefs

Cf. A143339, A143340, A143341.

Sequence in context: A184266 A000683 A352357 * A084270 A284195 A053677

Adjacent sequences: A143339 A143340 A143341 * A143343 A143344 A143345

Keyword

nonn

Author

Paul D. Hanna, Aug 09 2008

Status

approved