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 A140094 G.f. satisfies: A(x) = x/(1 - A(A(A(x)))). 4
 1, 1, 4, 25, 199, 1855, 19387, 221407, 2717782, 35455981, 487672243, 7029980797, 105732907498, 1653377947393, 26805765569863, 449568735630517, 7785116448484318, 138980739891821269, 2554369130466577138 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Paul D. Hanna, Table of n, a(n), n = 1..100. FORMULA G.f. A(x) satisfies: (1) A(x) = Series_Reversion(x - x*A(A(x))). (2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!. (3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ). Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x, then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ; thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x). G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with: A = 1 + x*A*C; B = A + x*B*D; C = B + x*C*E; D = C + x*D*F; E = D + x*E*G; ... EXAMPLE G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +... Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows. A_2(x) = 1 - Series_Reversion( A(x) ) / x; A_3(x) = 1 - x / A(x); A_4(x) = 1 - A(x) / A_2(x); A_5(x) = 1 - A_2(x) / A_3(x); A_6(x) = 1 - A_3(x) / A_4(x); ... where the iterations of A(x) begin: A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...; A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...; A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...; A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...; A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...; ... Iterations are also related by continued fractions: A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ; A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))). PROG (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))} CROSSREFS Cf. A140095, A088714. Cf. A088717, A091713, A120971, A139702. Sequence in context: A182953 A327074 A337167 * A284859 A144647 A215505 Adjacent sequences:  A140091 A140092 A140093 * A140095 A140096 A140097 KEYWORD nonn AUTHOR Paul D. Hanna, May 08 2008, May 20 2008 EXTENSIONS Name, formulas, and examples revised by Paul D. Hanna, Feb 03 2013 STATUS approved

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Last modified September 20 08:10 EDT 2021. Contains 347577 sequences. (Running on oeis4.)