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, 48277900432914176227, 937357175357395653301, 18679170546731484738259, 381722027611740836329630

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n), n = 1..100.

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(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-1) * A(A(x))^n / n! ).

(4) A^n(x) = A^(n-1) / (1 - A^(n+2)(x)).

(5) A^n(x) = 1 - A^(n-3)(x) / A^(n-2)(x).

(6) A(x) = 1 - A^(-2)(x) / A^(-1)(x) where -A^(-1)(-x) = x - x*A^2(-x) is the g.f. of A396102. - Paul D. Hanna, Jun 07 2026

Example

G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 + 221407*x^8 + 2717782*x^9 + 35455981*x^10 + ...
Iterations of A(x) are related as follows.
A^2(x) = 1 - A^(-1)(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 + ...;
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 of A(x) are also related by continued fractions
A(x) = x/(1 - A^2(x)/(1 - A^4(x)/(1 - A^6(x)/(1 - ...)))), and
A^2(x) = A(x)/(1 - A^3(x)/(1 - A^5(x)/(1 - A^7(x)/(1 - ...)))).
Also, A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
  A = x + A*C;
  B = A + B*D;
  C = B + C*E;
  D = C + D*F; ...
  resulting in B = A^2(x), C = A^3(x), D = A^4(x), etc.
The series reversion of A(x) begins
A^(-1)(x) = x - x^2 - 2*x^3 - 10*x^4 - 71*x^5 - 616*x^6 - 6119*x^7 - 67210*x^8 - 799307*x^9 + ... + (-1)^(n-1)*A396102(n)*x^n + ... also,
A^(-2)(x) = x - 2*x^2 - 2*x^3 - 11*x^4 - 80*x^5 - 701*x^6 - 6998*x^7 - 77075*x^8 - 918059*x^9 + ...
where A(x) = 1 - A^(-2)(x)/A^(-1)(x).

Prog

(PARI) {a(n) = my(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)))); polcoef(GF=A, n))}
{upto(n) = a(n); Vec(GF)}
upto(30)

Crossrefs

Cf. A396102, 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.

Entry revised by Paul D. Hanna, Jun 07 2026

Status

approved