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%I #8 Aug 03 2012 14:51:06
%S 1,1,1,5,21,125,825,6133,49925,439417,4142945,41544161,440710117,
%T 4924691541,57766255689,709205703565,9090541134373,121389729560633,
%U 1685431945085489,24289856880005441,362776874949660485,5606980244843123077,89560387072919814553
%N G.f. A(x) satisfies: A(A(x)) = x+x^2 + x*A(A(A(A(x)))).
%C Given g.f. A(x), A(A(x)) equals the g.f. of A213010.
%H Paul D. Hanna, <a href="/A213009/b213009.txt">Table of n, a(n) for n = 1..256</a>
%F a(n) == 1 (mod 4).
%e G.f.: A(x) = x + x^2 + x^3 + 5*x^4 + 21*x^5 + 125*x^6 + 825*x^7 +...
%e where
%e A(A(x)) = x + 2*x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 480*x^6 + 3296*x^7 +...
%e A(A(A(A(x)))) = x + 4*x^2 + 16*x^3 + 80*x^4 + 480*x^5 + 3296*x^6 +...
%o (PARI) {a(n)=local(A=x+x^2,B=x+2*x^2);for(i=1,n,B=x+x^2+x*subst(B,x,B+x*O(x^n)));
%o for(i=1,n,A=(A+subst(B,x,serreverse(A+x*O(x^n))))/2);polcoeff(A,n)}
%o for(n=1,31,print1(a(n),", "))
%Y Cf. A213010, A215115, A215117, A215119.
%K nonn
%O 1,4
%A _Paul D. Hanna_, Jun 01 2012
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