OFFSET
0,3
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 514*x^5 + 3444*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 1484*x^5 + 9520*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 3024*x^5 + 19240*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 5264*x^5 + 33800*x^6 +...
A(x)*A(-x) = 1 + 7*x^2 + 252*x^4 + 6496*x^6 + 308820*x^8 + 10966136*x^10 + 582452652*x^12 + 23322250960*x^14 + 1309365750212*x^16 +...
Note that A(x) = 1 + x*A(x)^4/(A(x)*A(-x)*A(I*x)*A(-I*x)) where
A(x)*A(-x)*A(I*x)*A(-I*x) = 1 + 455*x^4 + 590200*x^8 + 1124826664*x^12 + 2538673877080*x^16 + 6294363022919816*x^20 + 16568529053651321656*x^24 +...
Note also that a bisection of 1/A(x)^3 equals a bisection of 1/A(x)^4:
1/A(x)^3 = 1 - 3*x - 6*x^2 - 28*x^3 - 165*x^4 + 273*x^5 - 2292*x^6 +...
1/A(x)^4 = 1 - 4*x - 6*x^2 - 28*x^3 - 165*x^4 + 728*x^5 - 2292*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x*A^3/(subst(A, x, -x)*subst(A, x, I*x)*subst(A, x, -I*x))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=1, n+1, A=1+x*A^4*exp(-4*sum(m=1, n\4, x^(4*m)*polcoeff(log(A), 4*m))+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 14 2012
STATUS
approved