0,2

a(n)=(-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(2)=-2. G.f.: A(x) = (1+2*x+sqrt(1+4*x-4*x^2))/2.

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) = g.f. of A025225, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos Sep 07 2005

G.f.: 4x^2/(1+2x-sqrt(1+4x-4x^2)). - Michael Somos Sep 08 2005

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

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

A^1=[1,2],-2,4,-12,40,-144,544,-2128,8544,...

A^2=[1,4,0],0,-4,16,-64,256,-1040,4288,...

A^3=[1,6,6,-4],0,0,-8,48,-240,1120,-5088,...

A^4=[1,8,16,0,-8],0,0,0,-16,128,-768,...

A^5=[1,10,30,20,-20,-8],0,0,0,0,-32,...

A^6=[1,12,48,64,-12,-48,0],0,0,0,0,0,...

A^7=[1,14,70,140,56,-112,-56,16],0,0,0,...

A^8=[1,16,96,256,240,-128,-256,0,32],0,0,...

the main diagonal of which equals:

[x^n]A(x)^(n+1) = (n+1)*A009545(n+1) for n>=0.

(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, 2, if(n==2, -2, (-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n)))} (PARI) {a(n)=polcoeff( (1+2*x+sqrt(1+4*x-4*x^2+x^2*O(x^n)))/2, n)}

(PARI) a(n)=polcoeff((1+2*x+sqrt(1+4*x-4*x^2+x*O(x^n)))/2, n)

Cf. A100223, A071356, A009545.

a(n)=-(-1)^n*A025227(n), if n>1.

Adjacent sequences: A100235 A100236 A100237 this_sequence A100239 A100240 A100241

Sequence in context: A019225 A002840 A007181 this_sequence A098774 A009264 A048157

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 30 2004