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%I #5 Dec 12 2021 22:50:37
%S 1,1,3,14,83,574,4433,37274,335751,3204616,32159272,337383121,
%T 3684215369,41735965645,489173496890,5919292080009,73818803114991,
%U 947369389903293,12496580900037441,169248542357479944
%N 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 G.f. A(x) satisfies: A_n(x) = Sum_{k>=0} ( A_{n+k-1}(x) )^(k+1) ; e.g.:
%F A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + A(A(A(A(x))))^5 + ...;
%F A(A(x)) = A(x) + A(A(x))^2 + A(A(A(x)))^3 + A(A(A(A(x))))^4 + ...;
%F A(A(A(x))) = A(A(x)) + A(A(A(x)))^2 + A(A(A(A(x))))^3 + ...; ...
%F Series_Reversion(A(x)) = x - x^2 - A(x)^3 - A(A(x))^4 - A(A(A(x)))^5 - ...
%e G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 83*x^5 + 574*x^6 + ...;
%e A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 292*x^5 + 2201*x^6 + ...;
%e A(A(A(x))) = x + 3*x^2 + 15*x^3 + 96*x^4 + 715*x^5 + 5921*x^6 + ...;
%e A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 176*x^4 + 1464*x^5 + 13322*x^6 + ...; ...
%e Series_Reversion(A(x)) = x - x^2 - x^3 - 4*x^4 - 21*x^5 - 133*x^6 - 959*x^7 - ...
%e ILLUSTRATE DEFINITION:
%e A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + ... where:
%e A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...;
%e A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1692*x^7 + ...;
%e A(A(A(x)))^4 = x^4 + 12*x^5 + 114*x^6 + 1032*x^7 + 9367*x^8 + ...;
%e A(A(A(A(x))))^5 = x^5 + 20*x^6 + 280*x^7 + 3440*x^8 + ...;
%e A(A(A(A(A(x)))))^6 = x^6 + 30*x^7 + 585*x^8 + 9490*x^9 + ...; ...
%o (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))}
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 15 2009
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