A212910 G.f. satisfies: A(x) = x^2 - x + Series_Reversion(x - x*A(x)).

1, 1, 1, 4, 11, 35, 125, 445, 1699, 6668, 26935, 112111, 476674, 2072146, 9182141, 41406119, 189830984, 883549848, 4171598085, 19962224926, 96746007976, 474586282085, 2355104582435, 11817111373152, 59928222117495, 307045555880793, 1588825668984517

Offset: 2

Author

Paul D. Hanna, May 30 2012

Keywords

• nonn

Comments

Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:

G(x) = Series_Reversion(x - x*G(x)),

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

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

Examples

			G.f.: A(x) = x^2 + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 +...
The series reversion of x - x*A(x) begins:
x + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 + 445*x^9 +...
which equals x - x^2 + A(x).
The g.f. satisfies:
A(x) = x^2 + 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(1-x + A(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! +...
Related expansions:
d/dx x^2*A(x)^2/2! = 3*x^5 + 7*x^6 + 12*x^7 + 45*x^8 + 155*x^9 +...
d^2/dx^2 x^3*A(x)^3/3! = 12*x^7 + 45*x^8 + 110*x^9 + 418*x^10 +...
d^3/dx^3 x^4*A(x)^4/4! = 55*x^9 + 286*x^10 + 910*x^11 + 3640*x^12 +...
d^4/dx^4 x^5*A(x)^5/5! = 273*x^11 + 1820*x^12 + 7140*x^13 +...
...
d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A001764(n)*x^(2*n+1) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 2..375

Crossrefs

Cf. A088714, A212919, A212922, A212923.

Programs

PARI

{a(n)=local(A=x^2);for(i=1,n,A=x^2-x+serreverse(x-x*A +x*O(x^n)));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))

PARI

{Dx(n,F)=local(G=F);for(i=1,n,G=deriv(G));G}
{a(n)=local(A=x^2);for(i=1,n,A=x^2+sum(m=1,n,Dx(m-1,x^m*A^m/m!)+x*O(x^n)));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))

PARI

{Dx(n,F)=local(G=F);for(i=1,n,G=deriv(G));G}
{a(n)=local(A=x^2);for(i=1,n,A=x^2-x+x*exp(sum(m=1,n,Dx(m-1,x^(m-1)*A^m/m!)+x*O(x^n))));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:

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

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