%I #18 Jul 29 2018 14:34:52
%S 0,1,1,2,3,4,7,7,14,11,25,16,41,22,63,29,92,37,129,46,175,56,231,67,
%T 298,79,377,92,469,106,575,121,696,137,833,154,987,172,1159,191,1350,
%U 211,1561,232,1793,254,2047,277,2324,301,2625,326,2951,352,3303,379
%N a(n) = binomial(n-j-1,j) + binomial(n-j,j-1) with j= floor((n-1)/2).
%C The column m=1 in the array A175685, where the sum over the binomials reduces to only two terms.
%H Colin Barker, <a href="/A175686/b175686.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F a(n) = A057979(n+1) + binomial(n-j,j-1) with j = A004526(n-1), n>0.
%F From _Benedict W. J. Irwin_, Oct 31 2016: (Start)
%F G.f.: -(x^3 - x^2 - x)*(x^4 - x^2 + 1)/(x^2 - 1)^4.
%F E.g.f.: ((6*x + 3*x^2)*cosh(x) + (42 + 21*x + 6*x^2 + x^3)*sinh(x))/48.
%F a(n) = (42 + 20*n + 6*n^2 + n^3 + (-1)^n*(-42 + 20*n - 6*n^2 + n^3))/96. (End)
%F a(n) = 4*a(n-2)-6*a(n-4)+4*a(n-6)-a(n-8) for n>7. - _Colin Barker_, Oct 31 2016
%t Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 1, Floor[(
%t n - 1)/2]}], {n, 0, 30}]
%t CoefficientList[Series[-(x^3-x^2-x)(x^4-x^2+1)/(x^2-1)^4, {x, 0, 30}],x] (* _Benedict W. J. Irwin_, Oct 31 2016 *)
%t Table[(42+20n+6n^2+n^3+(-1)^n(-42+20n-6n^2+n^3))/96, {n, 0, 30}] (* _Benedict W. J. Irwin_, Oct 31 2016 *)
%t LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,1,1,2,3,4,7,7},60] (* _Harvey P. Dale_, Jul 29 2018 *)
%o (PARI) concat(0, Vec(x*(1+x-x^2)*(1-x^2+x^4)/((1-x)^4*(1+x)^4) + O(x^100))) \\ _Colin Barker_, Oct 31 2016
%Y Cf. A175685, A057979, A152271.
%K nonn,easy
%O 0,4
%A _Roger L. Bagula_, Dec 04 2010
%E More terms from _Colin Barker_, Oct 31 2016