A144005 E.g.f. A(x) satisfies: A(x) = 1 + Series_Reversion( Integral A(-x) dx ).

1, 1, 1, 2, 7, 33, 201, 1479, 12842, 127952, 1440989, 18070767, 249766088, 3769280801, 61654447712, 1085974748430, 20485430748783, 411839042136379, 8786499316562396, 198174104269740313, 4708919322491690592

Offset: 0

Author

Paul D. Hanna, Sep 07 2008

Keywords

• nonn

Comments

This is essentially the same sequence as A014623. - N. J. A. Sloane, Jul 24 2022

Compare the definition of e.g.f. A(x) to the trivial statement:

if F(x) = 1 + Series_Reversion( Integral F(-x)^2 dx ) then F(x) = 1/(1-x).

Examples

			E.g.f: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 33*x^5/5! +...
Let I(x) = Series_Reversion(A(x) - 1) = Integral A(-x) dx then
I(x) = x - x^2/2! + x^3/3! - 2*x^4/4! + 7*x^5/5! - 33*x^6/6! +...
		

Crossrefs

Cf. A014623; A144002.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+serreverse(intformal(subst(A,x,-x)^1)));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies: A(x) = 1 + Integral 1/A(1 - A(x)) dx. - Paul D. Hanna, Jul 10 2015