%I #10 Oct 08 2018 18:36:03

%S 1,4,4,2,4,2,4,4,2,4,4,2,2,4,4,2,2,4,4,3,4,3,2,4,1,2,4,2,3,1,4,2,4,3,

%T 1,4,4,4,2,2,2,3,3,2,3,2,2,4,1,4,2,2,1,4,3,3,3,1,1,3,3,4,4,3,3,3,3,1,

%U 4,4,3,2,4,2,2,2,1,3,4,2,3,3,1,4,2,3,1,1,3,3,4,2,4,3,1,4,3,2,1,1,1,2,1,4,4

%N Unique sequence of numbers {1,2,3,4} where g.f. A(x) satisfies A(x) = B(B(B(B(x)))) (4th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

%e G.f.: A(x) = x + 4*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + ...

%e then A(x) = B(B(B(B(x)))) where

%e B(x) = x + x^2 - 2*x^3 + 8*x^4 - 38*x^5 + 194*x^6 - 992*x^7 + ...

%e is the g.f. of A112109.

%o (PARI) {a(n,m=4)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

%Y Cf. A112109, A112104-A112107, A112110-A112127.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 27 2005