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

0,3

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.

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

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

Sequence in context: A099784 A365160 A082747 * A242796 A298527 A071288

Adjacent sequences: A127272 A127273 A127274 * A127276 A127277 A127278

Keyword

easy,sign

Author

Paul D. Hanna, Jun 07 2006

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

Status

approved