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%I #9 Feb 03 2013 19:25:42
%S 1,1,4,25,199,1855,19387,221407,2717782,35455981,487672243,7029980797,
%T 105732907498,1653377947393,26805765569863,449568735630517,
%U 7785116448484318,138980739891821269,2554369130466577138
%N G.f. satisfies: A(x) = x/(1 - A(A(A(x)))).
%H Paul D. Hanna, <a href="/A140094/b140094.txt">Table of n, a(n), n = 1..100.</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Series_Reversion(x - x*A(A(x))).
%F (2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!.
%F (3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ).
%F Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
%F then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
%F thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
%F G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
%F A = 1 + x*A*C;
%F B = A + x*B*D;
%F C = B + x*C*E;
%F D = C + x*D*F;
%F E = D + x*E*G; ...
%e G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
%e Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
%e A_2(x) = 1 - Series_Reversion( A(x) ) / x;
%e A_3(x) = 1 - x / A(x);
%e A_4(x) = 1 - A(x) / A_2(x);
%e A_5(x) = 1 - A_2(x) / A_3(x);
%e A_6(x) = 1 - A_3(x) / A_4(x); ...
%e where the iterations of A(x) begin:
%e A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...;
%e A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
%e A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...;
%e A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...;
%e A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...;
%e ...
%e Iterations are also related by continued fractions:
%e A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
%e A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).
%o (PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, subst(A, x, A)))); polcoeff(A, n))}
%Y Cf. A140095, A088714.
%Y Cf. A088717, A091713, A120971, A139702.
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 08 2008, May 20 2008
%E Name, formulas, and examples revised by _Paul D. Hanna_, Feb 03 2013
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