OFFSET
0,8
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.
FORMULA
a(n) = Sum_{k=0..floor(n/2)} ((C(n-k,k)/(n-2*k+1))*Sum_{i=0..k}(C(k,i)*C(n-k-i-1,n-2*k-i)*2^(n-2*k-i)*(-1)^(k-i)).
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = (2*x + x^3) * y^2 - (1 + x)^2 * y + 1. - Michael Somos, Jun 13 2010
G.f.: 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4)). - Michael Somos, Jun 13 2010
Conjecture: 2*n*(n+1)*a(n) +2*n*(n+1)*a(n-1) -3*(3*n-1)*(3*n-4)*a(n-2) +(61*n^2-191*n+36)*a(n-3) +6*(-2*n^2+2*n-1)*a(n-4) +2*(5*n-1)*(4*n-15)*a(n-5) +n*(n-5)*a(n-6) +(5*n-1)*(n-6)*a(n-7)=0. - R. J. Mathar, Jun 14 2016
EXAMPLE
G.f. = 1 - x^2 - x^3 - x^4 - x^5 + x^6 + 8*x^7 + 23*x^8 + 45*x^9 + 55*x^10 + ...
MATHEMATICA
CoefficientList[Series[2/(1+2*x+x^2+Sqrt[1-4*x+6*x^2+x^4]), {x, 0, 40}], x] (* G. C. Greubel, Sep 22 2018 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4 + x*O(x^n))), n))}; /* Michael Somos, Jun 13 2010 */
(PARI) x='x+O('x^50); Vec(2/(1+2*x+x^2+(1-4*x+6*x^2+x^4)^(1/2))) \\ Altug Alkan, Sep 23 2018
(Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2/(1+2*x+x^2+Sqrt(1-4*x+6*x^2+x^4)))); // G. C. Greubel, Sep 22 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, May 19 2010
STATUS
approved