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 A175686 a(n) = binomial(n-j-1,j) + binomial(n-j,j-1) with j= floor((n-1)/2). 3

%I

%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

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)