OFFSET
0,2
COMMENTS
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.
FORMULA
a(n)=-((4*n-6)*a(n-1)+20*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=3, a(3)=4. G.f.: A(x) = (1+4*x+sqrt(1+4*x+20*x^2))/2.
EXAMPLE
From the table of powers of A(x) (A100232), we see that
5^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,3],4,-8,0,64,-192,-128,...
A^2=[1,6,17],8,-32,64,64,-896,...
A^3=[1,9,39,75],12,-72,256,-384,...
A^4=[1,12,70,220,321],16,-128,640,...
A^5=[1,15,110,470,1165,1363],20,-200,...
A^6=[1,18,159,852,2895,5922,5777],24,...
PROG
(PARI) a(n)=if(n==0, 1, (5^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, 3, if(n==2, 4, -((4*n-6)*a(n-1)+20*(n-3)*a(n-2))/n)))
(PARI) a(n)=polcoeff((1+4*x+sqrt(1+4*x+20*x^2+x^2*O(x^n)))/2, n)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 29 2004
STATUS
approved