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%I #39 Sep 08 2022 08:46:04
%S 1,2,4,7,14,24,48,82,164,280,560,956,1912,3264,6528,11144,22288,38048,
%T 76096,129904,259808,443520,887040,1514272,3028544,5170048,10340096,
%U 17651648,35303296,60266496,120532992,205762688,411525376,702517760,1405035520,2398545664,4797091328,8189147136,16378294272,27959497216
%N Expansion of (1+2*x-x^3)/(1-4*x^2+2*x^4).
%C In general, a(n,j,m) = Sum_{r=1..m} (2^n*(1-(-1)^r)*cos(Pi*r/(m+1))^n*cot(Pi*r/(2*(m+1)))*sin(j*Pi*r/(m+1)))/(m+1) gives the number of paths of length n starting at the j-th node on the path graph P_m. Here we have the case m=7 and j=3. - _Herbert Kociemba_, Sep 17 2020
%H Shaun V. Ault and Charles Kicey, <a href="http://dx.doi.org/10.1016/j.disc.2014.05.020">Counting paths in corridors using circular Pascal arrays</a>, Discrete Math. 332 (2014), 45--54. MR3227977. See Fig. 5. - _N. J. A. Sloane_, Aug 04 2014
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-2).
%F G.f.: (1+x)*(1+x-x^2)/(1-4*x^2+2*x^4).
%F a(n) = Sum_{k=0..n} A216232(n-k,k).
%F a(n) = 4*a(n-2) - 2*a(n-4) for n>=4, a(0)=1, a(1)=2, a(2)=4, a(3)=7.
%F a(2*n) = A007070(n), a(2*n-1) = a(2*n)/2 = A007070(n)/2.
%F a(n)*a(n+1)-a(n-1)*a(n+2) = (1-(-1)^n)*2^floor(n/2-1) for n>0. - _Bruno Berselli_, Mar 22 2013
%F a(n) = Sum_{r=1..7} (2^n*(1-(-1)^r)*cos(Pi*r/8)^n*cot(Pi*r/16)*sin(3*Pi*r/8))/8. - _Herbert Kociemba_, Sep 17 2020
%t CoefficientList[Series[(1 + 2 x - x^3)/(1 - 4 x^2 + 2 x^4), {x, 0, 40}], x] (* _Bruno Berselli_, Mar 22 2013 *)
%t a[n_,j_,m_]:=Sum[(2^(n+1)Cos[Pi r/(m+1)]^n Cot[Pi r/(2(m+1))] Sin[j Pi r/(m+1)])/(m+1),{r,1,m,2}]
%t Table[a[n,3,7],{n,0,40}]//Round (* _Herbert Kociemba_, Sep 17 2020 *)
%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x-x^3)/(1-4*x^2+2*x^4))); // _Bruno Berselli_, Mar 22 2013
%o (Maxima) makelist(coeff(taylor((1+2*x-x^3)/(1-4*x^2+2*x^4), x, 0, n), x, n), n, 0, 40); /* _Bruno Berselli_, Mar 22 2013 */
%Y Cf. A007070, A077957, A204089, A216232.
%Y First differences are in A062113.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Mar 22 2013
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