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%I #28 Mar 22 2023 12:50:21
%S 1,0,2,2,6,12,26,60,130,300,672,1540,3514,8064,18552,42756,98802,
%T 228624,530024,1230372,2860000,6655792,15505932,36159552,84398626,
%U 197154984,460903796,1078251044,2524144224,5912535672,13857378300,32495267712
%N Expansion of 1/sqrt(1-4*x^2-4*x^3).
%C Diagonal sums of number triangle A115951.
%C Number of lattice paths from (0,0) to (n,n) using steps (2,1), (1,0), (1,2). - _Joerg Arndt_, Jul 05 2011
%C Diagonal of rational function 1/(1 - (x^2 + y^2 + x^3*y)). - _Seiichi Manyama_, Mar 22 2023
%H G. C. Greubel, <a href="/A115962/b115962.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..floor(n/2)} C(2*k,k)*C(k,n-2*k).
%F G.f.: Q(0), where Q(k) = 1 + 4*x*(x+x^2)*(4*k+1) / (4*k+2 - 4*x*(x+x^2)*(4*k+2)*(4*k+3) / (4*x*(x+x^2)*(4*k+3) + 4*(k+1) / Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Sep 14 2013
%F D-finite with recurrence: n*a(n) - 4*(n-1)*a(n-2) - 2*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Jan 14 2020
%p A115962 := proc(n)
%p option remember;
%p if n < 4 then
%p op(n+1,[1,0,2,2]);
%p else
%p 4*(n-1)*procname(n-2)+2*(2*n-3)*procname(n-3) ;
%p %/n ;
%p end if;
%p end proc:
%p seq(A115962(n),n=0..20) ; # _R. J. Mathar_, Jan 14 2020
%t CoefficientList[Series[1/Sqrt[1-4x^2-4x^3], {x, 0, 35}], x] (* or *) Table[Sum[Binomial[2k, k] Binomial[k, n-2k], {k, 0, Floor[n/2]}], {n, 0, 35}] (* _Michael De Vlieger_, Sep 03 2015 *)
%o (PARI) x = xx+O(xx^40); Vec(1/sqrt(1-4*x^2-4*x^3)) \\ _Michel Marcus_, Sep 03 2015
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1-4*x^2-4*x^3) )); // _G. C. Greubel_, May 06 2019
%o (Sage) (1/sqrt(1-4*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 06 2019
%Y CF. A006139, A361726.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 14 2006
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