A174399 Expansion of (1-x-x^2-sqrt(1-2x-5x^2+10x^3+x^4))/(2x^2).

1, -1, 1, -2, 2, -6, 5, -21, 14, -79, 43, -308, 147, -1221, 571, -4868, 2514, -19388, 12144, -76814, 61681, -302007, 318597, -1177274, 1640389, -4553897, 8333655, -17533572, 41583474, -67607944, 203455513, -263678119, 975780382

Offset

0,4

Comments

G.f. A(x) satisfies A(x)=1-2x+x*A(x)+x^2*A(x)+x^2*A(x)^2.

Hankel transform is the (essentially) (1,-1) Somos-4 sequence A174400.

Links

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

Formula

G.f.: (1-x-x^2-sqrt(1-2x-5x^2+10x^3+x^4))/(2x^2).

D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +5*(1-n)*a(n-2) +5*(2*n-5)*a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012

Recurrence verified using d.e. (x^5+10*x^4-5*x^3-2*x^2+x) y'' + (5*x^3-5*x^2-3*x+2) y' + 2*x^3-4*x^2+6*x-2 = 0 satisfied by the G.f. - Robert Israel, Jul 21 2019

Maple

f:= gfun:-rectoproc({(n+2)*a(n) -(2*n+1)*a(n-1) +5*(1-n)*a(n-2) +5*(2*n-5)*a(n-3) +(n-4)*a(n-4)=0, a(0)=1, a(1)=-1, a(2)=1, a(3)=-2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jul 21 2019

Mathematica

CoefficientList[Series[(1-x-x^2-Sqrt[1-2x-5x^2+10x^3+x^4])/(2x^2), {x, 0, 40}], x] (* Harvey P. Dale, Jan 28 2015 *)

Prog

(PARI) my(x='x+O('x^50)); Vec((1-x-x^2-sqrt(1-2*x-5*x^2+10*x^3+x^4))/(2*x^2)) \\ G. C. Greubel, Sep 22 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-x^2-Sqrt(1-2*x-5*x^2+10*x^3+x^4))/(2*x^2))); // G. C. Greubel, Sep 22 2018

Crossrefs

Cf. A174400.

Sequence in context: A111419 A360087 A306768 * A056881 A260322 A286540

Adjacent sequences: A174396 A174397 A174398 * A174400 A174401 A174402

Keyword

easy,sign

Author

Paul Barry, Mar 18 2010

Extensions

Corrected and extended by T. D. Noe, Apr 26 2010

Status

approved