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A153389 a(n) is the coefficient of x^n in the (n+1)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.

%I A153389 #2 Mar 30 2012 18:37:15
%S A153389 1,1,3,24,340,6950,185332,6102096,240066324,11049803253,586427736400,
%T A153389 35496295553400,2429241782433336,186578739919860938,
%U A153389 15978032482994396274,1516714259732732880752,158739955140174552148004
%N A153389 a(n) is the coefficient of x^n in the (n+1)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.
%e A153389 G.f.: A(x) = x + x^2 + 3*x^3 + 24*x^4 + 340*x^5 + 6950*x^6 +...
%e A153389 The initial coefficients of successive iterations of A(x) are as follows:
%e A153389 [1,1,3,24,340,6950,185332,6102096,240066324,11049803253,...];
%e A153389 [(1),2,8,64,866,17042,442452,14294645,554552353,25248014932,...];
%e A153389 [1,(3),15,126,1666,31676,799438,25293855,966179243,...];
%e A153389 [1,4,(24),216,2852,52800,1295784,40085834,1505339722,...];
%e A153389 [1,5,35,(340),4560,83030,1986288,60029146,2212932714,...];
%e A153389 [1,6,48,504,(6950),125770,2945660,86988543,3144212187,...];
%e A153389 [1,7,63,714,10206,(185332),4273850,123504913,4373929873,...];
%e A153389 [1,8,80,976,14536,267056,(6102096),173006484,6003091828,...];
%e A153389 [1,9,99,1296,20172,377430,8599692,(240066324),8167689120,...];
%e A153389 [1,10,120,1680,27370,524210,11981476,330711177,(11049803253),...];...
%e A153389 where the diagonal of terms in parenthesis form this sequence.
%e A153389 Some explicit expansions:
%e A153389 A(A(x)) = x + 2*x^2 + 8*x^3 + 64*x^4 + 866*x^5 + 17042*x^6 +...;
%e A153389 A(A(A(x))) = x + 3*x^2 + 15*x^3 + 126*x^4 + 1666*x^5 + 31676*x^6 +...;
%e A153389 A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 216*x^4 + 2852*x^5 + 52800*x^6 +...;
%o A153389 (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, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
%Y A153389 Cf. A119819.
%K A153389 nonn
%O A153389 1,3
%A A153389 _Paul D. Hanna_, Jan 04 2009