A210949 E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x))).

1, 1, 4, 29, 309, 4383, 78121, 1684706, 42801222, 1255919755, 41918624013, 1572257236114, 65619165625383, 3022617826829288, 152615633802149416, 8397224009015443509, 500957609480739613321, 32261529179806961067634, 2234133327582388824135291

Offset: 1

Author

Paul D. Hanna, Jul 22 2012

Keywords

• nonn

Comments

An unsigned version of A067146.

Equals row sums of triangle A277410.

Is this a duplicate of A014622, which is related to f(f(x))f'(x)=-1 ? - R. J. Mathar, May 13 2025

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 29*x^4/4! + 309*x^5/5! + 4383*x^6/6! +...
Let G(x) = Integral A(x) dx, then A(x) = x + G(A(x)) where
G(x) = x^2/2! + x^3/3! + 4*x^4/4! + 29*x^5/5! + 309*x^6/6! + 4383*x^7/7! +...
Also,
A(x) = x + G(x) + d/dx G(x)^2/2! + d^2/dx^2 G(x)^3/3! + d^3/dx^3 G(x)^4/4! +...
log(A(x)/x) = G(x)/x + d/dx G(x)^2/(2!*x) + d^2/dx^2 G(x)^3/(3!*x) + d^3/dx^3 G(x)^4/(4!*x) +...
By definition, A'(x) = 1/(1 - A(A(x))), where
A(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 101*x^4/4! + 1313*x^5/5! + 22235*x^6/6! + 466356*x^7/7! + 11710760*x^8/8! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..180

Crossrefs

Cf. A277410, A067146, A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Programs

Mathematica

m = 20; A[_] = 0;
Do[A[x_] = InverseSeries[Integrate[1 - A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x] * Range[0, m - 1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)

PARI

{a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=serreverse(x-G));n!*polcoeff(A, n)}

PARI

{a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=x+subst(G,x,A+x*O(x^n))); n!*polcoeff(A, n)}

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, G^m/m!))); n!*polcoeff(A, n)}

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, G^m/x/m!)+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

PARI

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

Formula

E.g.f. A(x) satisfies:

(1) A(x) = Series_Reversion( Integral 1 - A(x) dx ).

(2) A''(x) = 1 / ( (1 - A(A(x)))^3 * (1 - A(A(A(x)))) ).

Let G(x) = Integral A(x) dx with G(0)=0, then the e.g.f. A(x) satisfies:

(3) A(x) = x + G(A(x)) or, equivalently, A(x - G(x)) = x.

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

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

a(n) = Sum_{k=0..n-1} A277410(n,k).