OFFSET
0,2
COMMENTS
Hankel transform is (-2)^C(n+1,2).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1 + 3*x - sqrt(1 + 2*x + 9*x^2))/(2*x).
a(n) = Sum_{k=0..n} C(n+k,2k)*A000108(k)*(-3)^(n-k).
a(n) = 0^n - 2*Sum_{k=0..floor((n-1)/2)} C(n-1,2k)*A000108(k)(-1)^(n-k-1)*2^k.
a(n) = Sum_{k=0..n} A090181(n,k)*(-2)^k. - Philippe Deléham, Feb 02 2009
D-finite with recurrence (n+1)*a(n) + (2*n-1)*a(n-1) + 9*(n-2)*a(n-2) = 0. - R. J. Mathar, Oct 25 2012
a(n) = (-3)^n*hypergeometric([-n, n+1], [2], 1/3). - Peter Luschny, Sep 17 2014
EXAMPLE
G.f. = 1 - 2*x + 2*x^2 + 2*x^3 - 10*x^4 + 6*x^5 + 42*x^6 + ... - Michael Somos, Sep 18 2018
MAPLE
A152681_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := -2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A152681_list(28); # Peter Luschny, May 19 2011
MATHEMATICA
CoefficientList[Series[(1+3*x-Sqrt[1+2*x+9*x^2])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Sep 14 2018 *)
PROG
(Sage)
A152681 = lambda n : (-3)^n*hypergeometric([-n, n+1], [2], 1/3)
[round(A152681(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 17 2014
(PARI) x='x+O('x^30); Vec((1 +3*x -sqrt(1+2*x+9*x^2))/(2*x)) \\ G. C. Greubel, Sep 14 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 +3*x -Sqrt(1+2*x+9*x^2))/(2*x))); // G. C. Greubel, Sep 14 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Dec 10 2008
STATUS
approved