OFFSET
1,3
COMMENTS
Conjecture: all terms are odd.
Conjecture: a(n) = [1,3,3,1] repeating (mod 4) for n > 2.
Conjecture: [x^n] A(A(x)) == 0 (mod 4) for n > 2.
Conjecture: [x^n] ( A(x) - x*A(A(A(x))) ) == 2 (mod 4) for n > 2.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n has the following properties, where A^n(x) denotes the n-th iteration of A(x).
(1.a) x = A( x - A(x)*A^2(x) ).
(1.b) x = A( x - x*A^2(x) - A^2(x)^2*A^3(x) ).
(1.c) x = A( x - A(x)^2 - A(x)*A^3(x)*A^4(x) ).
(1.d) x = A^2( x - x*A(x) - A(x)*A^2(x) ).
(1.e) x = A^2( x-x^2 - A(x)*A^2(x) - x*A^2(x)*A^3(x) ).
(1.f) x = A^3( x-x^2 - x*A(x) - (1-x)*A(x)*A^2(x) ).
(2.a) A(x) = x + A^2(x)*A^3(x).
(2.b) A(x) = x + A^2(x)^2 + A^2(x)*A^4(x)*A^5(x).
(2.c) A(x) = (x + A^3(x)^2*A^4(x)) / (1 - A^3(x)).
(3.a) A^2(x) = A(x) + A^3(x)*A^4(x).
(3.b) A^2(x) = A(x) + A^3(x)^2 + A^3(x)*A^5(x)*A^6(x).
(3.c) A^2(x) = (x + A^3(x)*A^4(x)) / (1 - A^3(x)).
(3.d) A^2(x) = (A(x) + A^4(x)^2*A^5(x)) / (1 - A^4(x)).
(4.a) A^3(x) = A^2(x) + A^4(x)*A^5(x).
(4.b) A^3(x) = A^2(x) + A^4(x)^2 + A^4(x)*A^6(x)*A^7(x).
(4.c) A^3(x) = (x + A^4(x)*A^5(x)) / (1 - A^2(x) - A^4(x)).
(4.d) A^3(x) = (A(x) + A^4(x)*A^5(x)) / (1 - A^4(x)).
(4.e) A^3(x) = (A^2(x) + A^5(x)^2*A^6(x)) / (1 - A^5(x)).
(5.a) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n*A^2(x)^n / n!.
(5.b) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) ( A(x)^n*A^2(x)^n/x ) / n! ).
EXAMPLE
G.f. A(x) = x + x^2 + 5*x^3 + 39*x^4 + 383*x^5 + 4357*x^6 + 55057*x^7 + 754223*x^8 + 11028735*x^9 + 170346677*x^10 + ...
The second and third iterations of A(x) begin
A(A(x)) = x + 2*x^2 + 12*x^3 + 104*x^4 + 1100*x^5 + 13252*x^6 + 175476*x^7 + 2500788*x^8 + 37843956*x^9 + 602523524*x^10 + ...
A(A(A(x))) = x + 3*x^2 + 21*x^3 + 201*x^4 + 2291*x^5 + 29327*x^6 + 408841*x^7 + 6094257*x^8 + 95986995*x^9 + 1584499919*x^10 + ...
where A(x) = x + A(A(x)) * A(A(A(x))).
Let R(x) satisfy R(A(x)) = x, then
R(x) = x - x^2 - 3*x^3 - 19*x^4 - 165*x^5 - 1725*x^6 - 20463*x^7 - 266651*x^8 - 3741877*x^9 - 55811245*x^10 + ...
where R(x) = x - A(x)*A(A(x)).
Therefore, g.f. A(x) satisfies
A(x) = x + A(x)*A(A(x)) + d/dx A(x)^2*A(A(x))^2/2! + d^2/dx^2 A(x)^3*A(A(x))^3/3! + d^3/dx^3 A(x)^4*A(A(x))^4/4! + d^4/dx^4 A(x)^5*A(A(x))^5/5! + ...
Also, g.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B*C;
B = A + C*D;
C = B + D*E;
D = C + E*F; ...
resulting in B = A^2(x), C = A^3(x), D = A^4(x), etc.
PROG
(PARI) \\ By definition
\\ (note: 'truncate' is used here only to reduce runtime)
{a(n) = my(A=x, B); for(i=1, n, A=truncate(A)+x^2*O(x^i);
B = subst(A, x, A); A = x + B*subst(A, x, B) +x*O(x^n)); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)
(PARI) \\ Using series reversion
{a(n) = my(A=x); for(i=1, n, A = serreverse(x - A*subst(A, x, A) +x*O(x^n)) ); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 02 2026
STATUS
approved