A152681 [x^(n+1)]Reversion[x*(1-x)/(1-3*x)].

1, -2, 2, 2, -10, 6, 42, -102, -82, 782, -814, -3854, 12454, 5014, -98694, 142218, 472158, -1932258, -19038, 14816994, -27370410, -64159962, 334154442, -121279878, -2418497010, 5523511086, 8914677362, -61259567662, 44249714438

Offset: 0

Author

Paul Barry, Dec 10 2008

Keywords

• easy
• sign

Comments

Hankel transform is (-2)^C(n+1,2).

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A125695.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 +3*x -Sqrt(1+2*x+9*x^2))/(2*x))); // G. C. Greubel, Sep 14 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 *)

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

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

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