OFFSET
0,3
COMMENTS
More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..230
FORMULA
G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).
a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 2*A(x)^(2*n+2)/(n+1) for n>=1 (see comment).
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 312*x^4 + 3965*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 68*x^3 + 700*x^4 + 8794*x^5 + 126974*x^6+..
A(x/A(x)^2) = 1 + x + 2*x^2 + 9*x^3 + 68*x^4 + 700*x^5 + 8794*x^6 +...
A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2):
G(x) = 1 + x + 6*x^2 + 59*x^3 + 742*x^4 + 10877*x^5 + 177612*x^6 +...
G(x)^2 = 1 + 2*x + 13*x^2 + 130*x^3 + 1638*x^4 + 23946*x^5 +...
To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1),
form a table of coefficients in A(x)^(2*n+2) as follows:
A^4: [(1), 4, 22, 172, 1753, 21612, 306348, ...];
A^6: [1, (6), 39, 320, 3267, 39756, 554595, ...];
A^8: [1, 8, (60), 520, 5366, 64816, 892308, ...];
A^10: [1, 10, 85, (780), 8190, 98702, 1344920, ...];
A^12: [1, 12, 114, 1108, (11895), 143676, 1943488, ...];
A^14: [1, 14, 147, 1512, 16653, (202384), 2725541, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[2/2*(1), 2/3*(6), 2/4*(60), 2/5*(780), 2/6*(11895), 2/7*(202384), ...].
PROG
(PARI) {a(n)=local(F=1+x, G); for(i=0, n, G=serreverse(x/(F+x*O(x^n))^2)/x; F=1+x*G^2); polcoeff(F, n)}
(PARI) /* This sequence is generated when k=2, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=2, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 09 2008
STATUS
approved