OFFSET
0,3
FORMULA
Eigenvector: a(n) = Sum_{k=0..[n/2]} n!/((n-2k)!*k!*(k+1)!)*a(k), for n>=0, with a(0)=1. G.f. satisfies: A(x) = A(-x/(1-2*x))/(1-2*x)); i.e., 2nd inverse binomial transform equals A(-x). G.f. satisfies: A(x/(1-x))/(1-x)) = A(-x/(1-3*x))/(1-3*x). G.f. of inverse binomial transform: A(x/(1+x))/(1+x)) = B(x^2) where [x^n] B(x) = a(n)*C(2*n,n)/(n+1) = a(n)*A000108(n) and A000108=Catalan.
EXAMPLE
A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 96*x^6 +...
A(x/(1+x))/(1+x) = 1 + x^2 + 2*2*x^4 + 4*5*x^6 + 11*14*x^8 +...
+ a(n)*A000108(n)*x^(2n) +...
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, n!/((n-2*k)!*k!*(k+1)!)*a(k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 09 2006
STATUS
approved