OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..4099
FORMULA
A series quadrisection of A(x) equals 2*x^2*A(x^4)^6.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 5*x^4 + 14*x^5 + 12*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 22*x^4 + 54*x^5 + 88*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 25*x^3 + 57*x^4 + 144*x^5 + 299*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 44*x^3 + 117*x^4 + 316*x^5 + 756*x^6 +...
A(x)^6 = 1 + 6*x + 27*x^2 + 104*x^3 + 345*x^4 + 1080*x^5 + 3113*x^6 +...
A(x)^8 = 1 + 8*x + 44*x^2 + 200*x^3 + 782*x^4 + 2800*x^5 + 9252*x^6 +...
where the series bisections of A(x)^2 are:
[A(x)^2 - A(-x)^2]/2 = 2*x*A(x^2)^6 and
[A(x)^2 + A(-x)^2]/2 = A(x^2)^4 + x^2*A(x^2)^8.
The series bisections of A(x)^3 are:
[A(x)^3 - A(-x)^3]/2 = 3*x*A(x^2)^8 + x^3*A(x^2)^12 and
[A(x)^3 + A(-x)^3]/2 = A(x^2)^6 + 3*x^2*A(x^2)^10.
The series bisections of A(x)^4 are:
[A(x)^4 - A(-x)^4]/2 = 4*x*A(x^2)^10 + 4*x^3*A(x^2)^14 and
[A(x)^4 + A(-x)^4]/2 = A(x^2)^8 + 6*x^2*A(x^2)^12 + x^4*A(x^2)^16.
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=subst(A, x, x^2+x*O(x^n))^2+x*subst(A, x, x^2+x*O(x^n))^4); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 20 2010
EXTENSIONS
Edited by Paul D. Hanna, Apr 22 2010
STATUS
approved