A140094 G.f. satisfies: A(x) = x/(1 - A(A(A(x)))).

1, 1, 4, 25, 199, 1855, 19387, 221407, 2717782, 35455981, 487672243, 7029980797, 105732907498, 1653377947393, 26805765569863, 449568735630517, 7785116448484318, 138980739891821269, 2554369130466577138

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