login
A127275
Expansion of (sqrt(1-4x)-x)/(1-4x).
6
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
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
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