0,6

G.f.: A(x) = (1+x+sqrt(1-2*x+5*x^2))/2. G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-x-x^2)]). a(n) = ((2*n-3)*a(n-1) - 5*(n-3)*a(n-2))/n (for n>2), with a(0)=1, a(1)=0, a(2)=1.

Given g.f. A(x), then B(x)=A(x)-1+x series reversion is -B(-x). - Michael Somos Sep 07 2005

Given g.f. A(x) and C(x) = reversion of x+x^2, then B(x)=A(x)-1+x satis fies B(x)=x+C(x*B(x)). - Michael Somos Sep 07 2005

From the table of powers of A(x) (A100224), we see that

2^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:

A^1=[1,0],1,1,0,-2,-3,1,11,...

A^2=[1,0,2],2,1,-2,-5,-2,12,...

A^3=[1,0,3,3],3,0,-5,-6,6,...

A^4=[1,0,4,4,6],4,-2,-8,-3,...

A^5=[1,0,5,5,10,10],5,-5,-10,...

A^6=[1,0,6,6,15,18,17],6,-9,...

A^7=[1,0,7,7,21,28,35,28],7,...

A^8=[1,0,8,8,28,40,60,64,46],...

the main diagonal of which is A001610 = [0,2,3,6,10,17,...],

where Sum_{n>=1} A001610(n-1)/n*x^n = log((1-x)/(1-x-x^2)).

(PARI) {a(n)=if(n==0, 1, (2^n-1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n+x*O(x^k), k)))/n)} (PARI) {a(n)=if(n==0, 1, if(n==1, 0, if(n==2, 1, ((2*n-3)*a(n-1)-5*(n-3)*a(n-2))/n)))} (PARI) {a(n)=polcoeff((1+x+sqrt(1-2*x+5*x^2+x^2*O(x^n)))/2, n)}

Cf. A007440, A100224, A100225.

Adjacent sequences: A100220 A100221 A100222 this_sequence A100224 A100225 A100226

Sequence in context: A046222 A074307 A007440 this_sequence A129969 A104379 A107415

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2004