A104497 Expansion of sqrt(1-8x)/sqrt(1-4x).

1, -2, -10, -52, -282, -1596, -9412, -57640, -364922, -2376812, -15852204, -107821656, -745342500, -5221954776, -36997822536, -264620356944, -1907962439994, -13852652486220, -101186612941084, -743057485099384, -5482375128919820, -40620301416309128

Offset

0,2

Comments

Convolution of A098579 and A000984.

Hankel transform is (-2)^n*A002315(n).

Links

Table of n, a(n) for n=0..21.

Formula

Moment representation: a(n) = (1/Pi)*Integral_{x = 4..8} (x^n*sqrt((8 - x)*(x - 4))/(x*(4 - x)) dx + sqrt(2)*0^n;

Also a(n) = C(2*n, n) - Sum_{k = 1..n} 2^(k+1)*C(2*k-2, k-1)*C(2*n-2*k, n-k)/k.

From Peter Bala, Feb 03 2024: (Start)

a(n) = - 2*A104498(n) for n >= 1.

G.f: c(-x/(1 - 8*x)) / c(x/(1 - 4*x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

P-recursive: n*a(n) = (12*n - 14)*a(n-1) - 32*(n - 2)*a(n-2) with a(0) = 1 and a(1) = -2. (End)

Crossrefs

Cf. A000108, A104498.

Sequence in context: A344576 A360317 A074612 * A166694 A238796 A216340

Adjacent sequences: A104494 A104495 A104496 * A104498 A104499 A104500

Keyword

easy,sign

Author

Paul Barry, Mar 11 2005, Mar 07 2008

Status

approved