A144005 - OEIS

%I #6 Jul 24 2022 22:07:49

%S 1,1,1,2,7,33,201,1479,12842,127952,1440989,18070767,249766088,

%T 3769280801,61654447712,1085974748430,20485430748783,411839042136379,

%U 8786499316562396,198174104269740313,4708919322491690592

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

%C This is essentially the same sequence as A014623. - _N. J. A. Sloane_, Jul 24 2022

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

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

%F E.g.f. satisfies: A(x) = 1 + Integral 1/A(1 - A(x)) dx. - _Paul D. Hanna_, Jul 10 2015

%e E.g.f: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 33*x^5/5! +...

%e Let I(x) = Series_Reversion(A(x) - 1) = Integral A(-x) dx then

%e I(x) = x - x^2/2! + x^3/3! - 2*x^4/4! + 7*x^5/5! - 33*x^6/6! +...

%o (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)}

%Y Cf. A014623; A144002.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 07 2008