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%I #8 Aug 03 2012 14:53:58
%S 1,4,16,256,4864,111616,2983936,89743360,2970861568,106768629760,
%T 4125849419776,170207219286016,7454572671926272,345078981839552512,
%U 16822127738969128960,860944587541763325952,46137178395559050870784,2582843669636660403896320,150735442996358913332346880
%N G.f. satisfies: A(x) = x + 3*x^2 + x*A(A(A(A(x)))).
%C The (1/4)-iteration of the g.f. equals an integer series (A215117).
%e G.f.: A(x) = x + 4*x^2 + 16*x^3 + 256*x^4 + 4864*x^5 + 111616*x^6 + 2983936*x^7 +...
%e where
%e A(A(A(A(x)))) = x + 16*x^2 + 256*x^3 + 4864*x^4 + 111616*x^5 + 2983936*x^6 +...
%e Related expansions.
%e Let D(D(D(D(x)))) = A(x), then D(x) is an integer series where:
%e D(x) = x + x^2 + x^3 + 49*x^4 + 721*x^5 + 17281*x^6 + 452065*x^7 +...
%e where the coefficients of D(x) are congruent to 1 modulo 48.
%o (PARI) {a(n)=local(A=x+4*x^2); for(i=1,n,A=x+3*x^2+x*subst(A,x,subst(A,x,subst(A,x,A+x*O(x^n))))); polcoeff(A, n)}
%o for(n=1, 31, print1(a(n), ", "))
%Y Cf. A215117, A213010, A215114, A215116, A215118.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 03 2012