A127275 - OEIS

%I #18 Jan 16 2023 11:15:29

%S 1,1,2,4,6,-4,-100,-664,-3514,-16916,-77388,-343144,-1490148,-6376616,

%T -26992264,-113317936,-472661434,-1961361076,-8104733884,-33374212936,

%U -137031378124,-561253753336,-2293947547384,-9358755316816,-38121140494564,-155064370272904

%N Expansion of (sqrt(1-4x)-x)/(1-4x).

%C Hankel transform is A127276.

%C The second self-composition of the g.f. G(x) of A120009 is G(G(x)) = (sqrt(1-4x)-x)/(1-4x) - 1.

%H Robert Israel, <a href="/A127275/b127275.txt">Table of n, a(n) for n = 0..1653</a>

%F a(n) = C(2n,n) - 4^(n-1) + 0^n/4. - _Paul Barry_, Jan 10 2007

%F 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

%F 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

%e A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 - 4*x^5 - 100*x^6 - 664*x^7 + ...

%p S:= series((sqrt(1-4*x)-x)/(1-4*x),x,31):

%p seq(coeff(S,x,i),i=0..30); # _Robert Israel_, Jan 15 2023

%o (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)}

%Y Cf. A120009, A120012 (3rd self-composition); A000108 (Catalan).

%K easy,sign

%O 0,3

%A _Paul D. Hanna_, Jun 07 2006

%E Definition revised by _Paul Barry_, Jan 10 2007

%E Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_ and _Max Alekseyev_