0,3

G.f.: A(x) = 2*x+sqrt(1-2*x+5*x^2).

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

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

A^1=[1,1],2,2,0,-4,-6,2,22,30,-26,...

A^2=[1,2,5],8,8,0,-16,-24,8,88,120,...

A^3=[1,3,9,19],30,30,2,-54,-84,20,288,...

A^4=[1,4,14,36,73],112,112,16,-176,-288,32,...

A^5=[1,5,20,60,145,281],420,420,90,-570,-988,...

A^6=[1,6,27,92,255,582,1085],1584,1584,440,-1848,...

A^7=[1,7,35,133,413,1071,2331,4201],6006,6006,2002,...

A^8=[1,8,44,184,630,1816,4460,9320,16305],22880,22880,...

the main diagonal of which is:

[x^n]A(x)^(n+1) = (n+1)*A081696(n) for n>=0.

(PARI) {a(n)=if(n==0, 1, (4^n/2-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(2*x+sqrt(1-2*x+5*x^2+x^2*O(x^n)), n)}

Cf. A100223, A057083, A007440, A081696.

Sequence in context: A073469 A086882 A168587 this_sequence A072690 A108520 A099087

Adjacent sequences: A100237 A100238 A100239 this_sequence A100241 A100242 A100243

sign

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