OFFSET
0,2
COMMENTS
G.f. = square root of weight enumerator of [4,3,2] even weight code.
a(n) gives the row sums of the coefficient array for the family Gegenbauer_C(n,-1/2,-2x-1). - Paul Barry, Apr 20 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
FORMULA
From Paul Barry, Apr 20 2009: (Start)
a(n) = (-1)^n*Sum_{k=0..n} C(n+k-2,n-k)*C(2*k,k)/(1-2*k).
a(n) = (-1)^n*Sum_{k=0..n} C(n+k-2,n-k)*A002420(k). (End)
G.f.: G(0)/2, where G(k)= 1 + 1/( 1 - x*(6+x)*(2*k-1)/(x*(6+x)*(2*k-1) - 2*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) ~ -(-1)^n * 2^(1/4) * (1 + sqrt(2))^(2*n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 12 2017
D-finite with recurrence: n*a(n) +3*(2*n-3)*a(n-1) +(n-3)*a(n-2) = 0. - R. J. Mathar, Jan 25 2020
EXAMPLE
G.f.: 1 + 3*x - 4*x^2 + 12*x^3 - 44*x^4 + 180*x^5 - 788*x^6 + 3612*x^7 - ...
MATHEMATICA
CoefficientList[Series[Sqrt[1+6x+x^2], {x, 0, 30}], x] (* Harvey P. Dale, Jun 30 2017 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( Sqrt(1+6*x+x^2) )); // G. C. Greubel, Nov 08 2025
(SageMath)
def A109771_list(prec):
P.<x>= PowerSeriesRing(QQ, prec)
return P( sqrt(1+6*x+x^2) ).list()
print(A109771_list(41)) # G. C. Greubel, Nov 08 2025
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane and Nadia Heninger, Aug 13 2005
STATUS
approved