A127275 Expansion of (sqrt(1-4x)-x)/(1-4x).

1, 1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, -343144, -1490148, -6376616, -26992264, -113317936, -472661434, -1961361076, -8104733884, -33374212936, -137031378124, -561253753336, -2293947547384, -9358755316816, -38121140494564, -155064370272904

Offset: 0

Author

Paul D. Hanna, Jun 07 2006

Keywords

• easy
• sign

Comments

Hankel transform is A127276.

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..1653

Crossrefs

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

Programs

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

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)}

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

Extensions

Definition revised by Paul Barry, Jan 10 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar and Max Alekseyev