OFFSET
0,3
COMMENTS
LINKS
Robert Israel, Table of n, a(n) for n = 0..1653
FORMULA
a(n) = C(2n,n) - 4^(n-1) + 0^n/4. - Paul Barry, Jan 10 2007
Conjecture: n*a(n) + 2*(-4*n+3)*a(n-1) + 8*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 26 2012
Conjecture verified using the differential equation (4*x-1)^2 * g'(x) + (8*x-2)*g(x) + 1 - 2*x = 0 satisfied by the g.f. - Robert Israel, Jan 15 2023
EXAMPLE
A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 - 4*x^5 - 100*x^6 - 664*x^7 + ...
MAPLE
S:= series((sqrt(1-4*x)-x)/(1-4*x), x, 31):
seq(coeff(S, x, i), i=0..30); # Robert Israel, Jan 15 2023
PROG
(PARI) {a(n)=local(k=2, x=X+X^3*O(X^n)); polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2, n, X)}
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul D. Hanna, Jun 07 2006
EXTENSIONS
Definition revised by Paul Barry, Jan 10 2007
STATUS
approved