A212923 G.f. satisfies: A(x) = x^2 + Series_Reversion(x - x*A(x)).

1, 2, 4, 19, 111, 734, 5338, 41839, 348827, 3065255, 28199803, 270253498, 2687629926, 27652068276, 293627150268, 3211604669731, 36124424800797, 417294625090201, 4944772338009206, 60045368928594948, 746560751627818906, 9496624640844863631, 123507266690219103213

Offset: 1

Author

Paul D. Hanna, May 31 2012

Keywords

• nonn

Comments

This is an application of the more general formula given by:

if G(x) = Series_Reversion(x - x*F(x)), with F(0)=0, then

(1) G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*F(x)^n/n!,

(2) G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*F(x)^n/n! );

here F(x) = A(x) and G(x) = A(x) - x^2.

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
The series reversion of x - x*A(x) begins:
x + x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
which equals A(x) - x^2.
The g.f. A(x) satisfies:
A(x) - x^2 = x + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(A(x)/x - x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..365

Crossrefs

Cf. A212922, A088714, A212910, A212919.

Programs

PARI

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

PARI

{Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x+x^2+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

PARI

{Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x+x^2); for(i=1, n, A=x^2+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:

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

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