OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n), n = 1..100.
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(x))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*C;
B = A + x*B*D;
C = B + x*C*E;
D = C + x*D*F;
E = D + x*E*G; ...
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion( A(x) ) / x;
A_3(x) = 1 - x / A(x);
A_4(x) = 1 - A(x) / A_2(x);
A_5(x) = 1 - A_2(x) / A_3(x);
A_6(x) = 1 - A_3(x) / A_4(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...;
A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...;
A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...;
A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...;
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).
PROG
(PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, subst(A, x, A)))); polcoeff(A, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 08 2008, May 20 2008
EXTENSIONS
Name, formulas, and examples revised by Paul D. Hanna, Feb 03 2013
STATUS
approved