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%I #19 Jan 30 2020 21:29:15
%S 1,3,-4,12,-44,180,-788,3612,-17116,83172,-412196,2075436,-10586892,
%T 54595476,-284157492,1490774076,-7875206076,41854313412,-223636052036,
%U 1200637707852,-6473448634348,35037238641780,-190299310403924,1036863750837852,-5665846701859484
%N G.f.: sqrt(1+6*x+x^2).
%C G.f. = square root of weight enumerator of [4,3,2] even weight code.
%C a(n) gives the row sums of the coefficient array for the family Gegenbauer_C(n,-1/2,-2x-1). [From _Paul Barry_, Apr 20 2009]
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%F a(n)=(-1)^n*sum{k=0..n, C(n+k-2,n-k)*C(2k,k)/(1-2k)}=(-1)^n*sum{k=0..n, C(n+k-2,n-k)*A002420(k)}; [From _Paul Barry_, Apr 20 2009]
%F 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
%F a(n) ~ -(-1)^n * 2^(1/4) * (1 + sqrt(2))^(2*n-1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 12 2017
%F 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
%e 1+3*x-4*x^2+12*x^3-44*x^4+180*x^5-788*x^6+3612*x^7-...
%t CoefficientList[Series[Sqrt[1+6x+x^2],{x,0,30}],x] (* _Harvey P. Dale_, Jun 30 2017 *)
%K sign
%O 0,2
%A _N. J. A. Sloane_ and _Nadia Heninger_, Aug 13 2005