Search: id:A154757
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%I A154757
%S A154757 1,1,3,14,83,574,4433,37274,335751,3204616,32159272,337383121,
%T A154757 3684215369,41735965645,489173496890,5919292080009,73818803114991,
%U A154757 947369389903293,12496580900037441,169248542357479944
%N A154757 G.f. A(x) = Sum_{n>=0} A_n(x)^(n+1) where A_{n+1}(x) = A_n( A(x) ) with 
               A_0(x)=x and A(0)=0.
%F A154757 G.f. A(x) satisfies: A_n(x) = Sum_{k>=0} ( A_{n+k-1}(x) )^(k+1) ; eg.:
%F A154757 A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + A(A(A(A(x))))^5 +...;
%F A154757 A(A(x)) = A(x) + A(A(x))^2 + A(A(A(x)))^3 + A(A(A(A(x))))^4 +...;
%F A154757 A(A(A(x))) = A(A(x)) + A(A(A(x)))^2 + A(A(A(A(x))))^3 +...; ...
%F A154757 Series_Reversion(A(x)) = x - x^2 - A(x)^3 - A(A(x))^4 - A(A(A(x)))^5 
               -...
%e A154757 G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 83*x^5 + 574*x^6 +...;
%e A154757 A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 292*x^5 + 2201*x^6 +...;
%e A154757 A(A(A(x))) = x + 3*x^2 + 15*x^3 + 96*x^4 + 715*x^5 + 5921*x^6 +...;
%e A154757 A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 176*x^4 + 1464*x^5 + 13322*x^6 +...; 
               ...
%e A154757 Series_Reversion(A(x)) = x - x^2 - x^3 - 4*x^4 - 21*x^5 - 133*x^6 - 959*x^7 
               -...
%e A154757 ILLUSTRATE DEFINITION:
%e A154757 A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 +... where:
%e A154757 A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 +...;
%e A154757 A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1692*x^7 +...;
%e A154757 A(A(A(x)))^4 = x^4 + 12*x^5 + 114*x^6 + 1032*x^7 + 9367*x^8 +...;
%e A154757 A(A(A(A(x))))^5 = x^5 + 20*x^6 + 280*x^7 + 3440*x^8 +...;
%e A154757 A(A(A(A(A(x)))))^6 = x^6 + 30*x^7 + 585*x^8 + 9490*x^9 +...; ...
%o A154757 (PARI) {a(n)=local(A=x,G,S);if(n<1,0, for(j=1,n,G=x;S=x;for(i=2,n+1,G=subst(A,
               x,G+x*O(x^n));S=S+G^i);A=S);polcoeff(A+x*O(x^n),n))}
%Y A154757 Sequence in context: A020104 A103467 A121687 this_sequence A074535 A005700 
               A088717
%Y A154757 Adjacent sequences: A154754 A154755 A154756 this_sequence A154758 A154759 
               A154760
%K A154757 nonn
%O A154757 1,3
%A A154757 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 15 2009

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