0,2

G.f.: A(x) = (1+3*x+sqrt(1+6*x-3*x^2))/2.

Given g.f. A(x), then B(x)=A(x)-1-2x series reversion is -B(-x). - Michael Somos Sep 07 2005

Given g.f. A(x) and C(x) = g.f. of A025226, then B(x)=A(x)-1-2x satisfies B(x)=x-C(x*B(x)). - Michael Somos Sep 07 2005

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

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

A^1=[1,3],-3,9,-36,162,-783,3969,-20817,...

A^2=[1,6,3],0,-9,54,-297,1620,-8910,49572,...

A^3=[1,9,18,0],0,0,-27,243,-1701,10935,...

A^4=[1,12,42,36,-9],0,0,0,-81,972,-8262,...

A^5=[1,15,75,135,45,-27],0,0,0,0,-243,...

A^6=[1,18,117,324,324,0,-54],0,0,0,0,0,...

A^7=[1,21,168,630,1071,567,-189,-81],0,0,0,...

A^8=[1,24,228,1080,2610,2808,540,-648,-81],0,0,...

the main diagonal of which is:

[x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.

(PARI) {a(n)=if(n==0, 1, (3^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)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2, n)}

Cf. A100226, A100239, A057083.

Sequence in context: A010098 A029857 A032086 this_sequence A038080 A115564 A122961

Adjacent sequences: A100236 A100237 A100238 this_sequence A100240 A100241 A100242

sign

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