0,3

More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.

G.f.: (1+2*x+sqrt(1+8*x^2))/2. G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-2*x-x^2)]). a(n) = -8*(n-3)*a(n-2)/n for n>2, with a(0)=1, a(1)=1, a(2)=2. a(2*n) = 2^n*(-1)^(n-1)*A000108(n-1), a(2*n+1)=0, for n>=1, where A000108=Catalan.

From the table of powers of A(x) (A100226), we see that

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

A^1=[1,1],2,0,-4,0,16,0,-80,...

A^2=[1,2,5],4,-4,-8,16,32,-80,...

A^3=[1,3,9,13],6,-12,-4,48,0,...

A^4=[1,4,14,28,33],8,-24,16,80,...

A^5=[1,5,20,50,85,81],10,-40,60,..

A^6=[1,6,27,80,171,246,197],12,-60,...

the main diagonal of which is A100227 = [1,5,13,33,81,197,477,...],

where Sum_{n>=1} A100227(n)/n*x^n = log((1-x)/(1-2*x-x^2).

(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)=if(n==0, 1, if(n==1, 1, if(n==2, 2, -8*(n-3)*a(n-2)/n)))} (PARI) {a(n)=polcoeff((1+2*x+sqrt(1+8*x^2+x^2*O(x^n)))/2, n)}

Cf. A100226, A100227, A000108, A025225, A100223, A100228.

Sequence in context: A056946 A111757 A022896 this_sequence A007420 A019219 A019139

Adjacent sequences: A100222 A100223 A100224 this_sequence A100226 A100227 A100228

sign

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