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A153849 a(n) is the coefficient of x^n in the (n+2)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.

%I A153849 #2 Mar 30 2012 18:37:15
%S A153849 1,1,4,40,675,16023,488264,18150408,798091737,40691618165,
%T A153849 2372928554920,156676126511294,11619829396974162,961567152709817189,
%U A153849 88268869225674860344,8941327301073770038304,994670849523023025795108
%N A153849 a(n) is the coefficient of x^n in the (n+2)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.
%e A153849 G.f.: A(x) = x + x^2 + 4*x^3 + 40*x^4 + 675*x^5 + 16023*x^6 +...
%e A153849 Coefficients of successive iterations of A(x) are as follows:
%e A153849 [1,1,4,40,675,16023,488264,18150408,798091737,40691618165,...];
%e A153849 [1,2,10,101,1658,38327,1144412,41900873,1821202358,...];
%e A153849 [(1),3,18,189,3063,69204,2024472,72937181,3130859853,...];
%e A153849 [1,(4),28,310,5028,111648,3202916,113479712,4806652072,...];
%e A153849 [1,5,(40),470,7715,169475,4777592,166446886,6951781033,...];
%e A153849 [1,6,54,(675),11310,247443,6875376,235657813,9700212066,...];
%e A153849 [1,7,70,931,(16023),351372,9658544,326079187,13225766705,...];
%e A153849 [1,8,88,1244,22088,(488264),13331864,444121464,17753548560,...];
%e A153849 [1,9,108,1620,29763,666423,(18150408),597989364,23574131721,...];
%e A153849 [1,10,130,2065,39330,895575,24428084,(798091737),31060981566,...];
%e A153849 [1,11,154,2585,51095,1186988,32546888,1057515833,(40691618165),...];
%e A153849 where the diagonal of terms in parenthesis form this sequence.
%e A153849 Some explicit expansions:
%e A153849 A(A(x)) = x + 2*x^2 + 10*x^3 + 101*x^4 + 1658*x^5 + 38327*x^6 +...;
%e A153849 A(A(A(x))) = x + 3*x^2 + 18*x^3 + 189*x^4 + 3063*x^5 + 69204*x^6 +...;
%e A153849 A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 310*x^4 + 5028*x^5 + 111648*x^6 +...
%o A153849 (PARI) {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
%Y A153849 Cf. A119819, A153389.
%K A153849 nonn
%O A153849 1,3
%A A153849 _Paul D. Hanna_, Jan 04 2009