OFFSET
1,3
COMMENTS
Conjecture: a(n) == 1 (mod 3) for n >= 1. - Paul D. Hanna, Dec 01 2024
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f.: A(x) = A(A(x))/(1 + A(A(A(A(A(x)))))).
G.f.: A(x) = Series_Reversion[ x/(1 + A(A(A(x)))) ].
G.f.: A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+3)) for n>0 with F(x,0)=1; further, x*F(x,n) is the n-th iteration of A(x).
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n} A_{3*k}(x), where A_n(x) denotes the n-th iteration of A(x) with A_0(x)=x.
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x*D;
B = A*(1 + x*E);
C = B*(1 + x*F);
D = C*(1 + x*G);
E = D*(1 + x*H); ...
The solution to the variables in the system of equations are:
A=A(x)/x, B=A(A(x))/x, C=A(A(A(x)))/x, D=A(A(A(A(x))))/x, etc.,
where iterations begin:
A(x) = x + x^2 + 4*x^3 + 28*x^4 + 262*x^5 + 2944*x^6 + 37666*x^7 +...
A(A(x)) = x + 2*x^2 + 10*x^3 + 77*x^4 + 760*x^5 + 8846*x^6 + 116140*x^7 +...
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1608*x^5 + 19566*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 262*x^4 + 2944*x^5 + 37666*x^6 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 40*x^3 + 410*x^4 + 4930*x^5 + 66530*x^6 +...
A(A(A(A(A(A(x)))))) = x + 6*x^2 + 54*x^3 + 603*x^4 + 7752*x^5 + 110484*x^6 +...
ALTERNATE GENERATING METHOD.
The g.f. A(x) equals the sum of products of {3*k}-iterations of A(x):
A(x) = x + x*A_3(x) + x*A_3(x)*A_6(x) + x*A_3(x)*A_6(x)*A_9(x) + x*A_3(x)*A_6(x)*A_9(x)*A_12(x) +...+ Product_{k=0..n} A_{3*k}(x) +...
where A_n(x) = A_{n-1}(A(x)) is the n-th iteration of A(x) with A_0(x)=x.
Related expansions.
x*A_3(x) = x^2 + 3*x^3 + 18*x^4 + 153*x^5 + 1608*x^6 + 19566*x^7 +...
x*A_3(x)*A_6(x) = x^3 + 9*x^4 + 90*x^5 + 1026*x^6 + 13059*x^7 +...
x*A_3(x)*A_6(x)*A_9(x) = x^4 + 18*x^5 + 279*x^6 + 4320*x^7 +...
x*A_3(x)*A_6(x)*A_9(x)*A_12(x) = x^5 + 30*x^6 + 675*x^7 +...
MATHEMATICA
Nest[x + x (# /. x -> # /. x -> # /. x -> #) &, O[x], 30][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)
PROG
(PARI) /* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* A(x) results from nested iterations of shifted series: */
(PARI) {a(n)=local(A=x); for(i=1, n, A=x+x*ITERATE(4, A, n)); polcoeff(A, n)}
(PARI) {a(n)=local(A=x); for(i=1, n, A=x+x*sum(m=1, n, prod(k=1, m, ITERATE(3*k, A, n)))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2011
STATUS
approved