1,3

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+3}(x)] ;

thus A_{n}(x) = 1 - A_{n-4}(x) / A_{n-3}(x).

G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:

A = 1 + xAD;

B = A + xBE;

C = B + xCF;

D = C + xDG;

E = D + xEH; ...

G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 437*x^5 + 5513*x^6 + 78477*x^7 +...

Self-compositions A_{n+1}(x) = A( A_{n}(x) ) are related as follows.

A_2(x) = 1 - Series_Reversion[A_2(x)]/Series_Reversion[A(x)] where

A_2(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1220*x^5 + 16028*x^6 +...

A_3(x) = 1 - Series_Reversion[A(x)]/x where

A_3(x) = x + 3*x^2 + 21*x^3 + 207*x^4 + 2489*x^5 + 34259*x^6 +...

A_4(x) = 1 - x/A(x) where

A_4(x) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4408*x^5 + 63776*x^6 +...

A_5(x) = 1 - A(x)/A_2(x) where

A_5(x) = x + 5*x^2 + 45*x^3 + 525*x^4 + 7165*x^5 + 109125*x^6 +...

A_6(x) = 1 - A_2(x)/A_3(x) where

A_6(x) = x + 6*x^2 + 60*x^3 + 756*x^4 + 10972*x^5 + 175948*x^6 +...

A_7(x) = 1 - A_3(x)/A_4(x) where

A_7(x) = x + 7*x^2 + 77*x^3 + 1043*x^4 + 16065*x^5 + 271103*x^6 +...

A_8(x) = 1 - A_4(x)/A_5(x) where

A_8(x) = x + 8*x^2 + 96*x^3 + 1392*x^4 + 22704*x^5 + 402784*x^6 +...

Self-compositions are also related by continued fractions:

A(x) = x/(1 - A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 -...)))) ;

A_2(x) = A(x)/(1 - A_4(x)/(1 - A_7(x)/(1 - A_10(x)/(1 -...)))) ;

A_3(x) = A_2(x)/(1 - A_5(x)/(1 - A_8(x)/(1 - A_11(x)/(1 -...)))).

(PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, B=subst(A, x, A); A=x/(1-subst(B, x, B))); polcoeff(A, n))}

Cf. A140094, A088714.

Cf. A088717, A091713, A120971, A139702.

Sequence in context: A058475 A064087 A081215 this_sequence A083073 A115257 A047735

Adjacent sequences: A140092 A140093 A140094 this_sequence A140096 A140097 A140098

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), May 08 2008, May 20 2008