A190761 Expansion of g.f. A(x) satisfying A(x) = x + A(A(x))^2 - A(A(x))^3.

1, 1, 3, 14, 84, 592, 4670, 40108, 368670, 3586321, 36632763, 390694000, 4332131804, 49777965585, 591173511887, 7241437905916, 91331043654080, 1184322726542850, 15770586926609276, 215423253906689779, 3015794930248824111, 43233248160139146114

Offset: 1

Author

Paul D. Hanna, May 18 2011

Keywords

• nonn

Comments

Compare to a formula for a g.f. of the Catalan numbers (A000108):

C(x) = x + C(x)*C(C(x)) - C(x)*C(C(x))^2 where C(x) = (1-sqrt(1-4*x))/2.

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + 40108*x^8 + 368670*x^9 + 3586321*x^10 + ...
Related series.
A(x) = x + A(A(x))^2 - A(A(x))^3 where
A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 294*x^5 + 2244*x^6 + 18888*x^7 + ...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 828*x^6 + 6368*x^7 + ...
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1698*x^7 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A368562, A213591.

Programs

PARI

{a(n) = my(A=x+x^2); for(i=1,n, A = serreverse(x - A^2 + A^3 +x*O(x^n))); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

PARI

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A= x + sum(m=1, n, Dx(m-1, A^(2*m)*(1 - A)^m )/m!) +x*O(x^n)); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 30 2023

Formula

From Paul D. Hanna, Dec 30 2023: (Start)

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

(1) A(x) = x + A(A(x))^2 - A(A(x))^3.

(2) x = A( x - A(x)^2 + A(x)^3 ).

(3) x = A(A( x - x^2 + x^3 - A(x)^2 + A(x)^3 )).

(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) * (1 - A(x))^n / n!.

(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)*(1 - A(x))^n/x / n! ).

(6) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:

A = x + B^2 - B^3;

B = A + C^2 - C^3;

C = B + D^2 - D^3;

D = C + E^2 - E^3; ...

where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.

... (End)