OFFSET
0,4
COMMENTS
The column m=1 in the array A175685, where the sum over the binomials reduces to only two terms.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).
FORMULA
From Benedict W. J. Irwin, Oct 31 2016: (Start)
G.f.: -(x^3 - x^2 - x)*(x^4 - x^2 + 1)/(x^2 - 1)^4.
E.g.f.: ((6*x + 3*x^2)*cosh(x) + (42 + 21*x + 6*x^2 + x^3)*sinh(x))/48.
a(n) = (42 + 20*n + 6*n^2 + n^3 + (-1)^n*(-42 + 20*n - 6*n^2 + n^3))/96. (End)
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
MATHEMATICA
Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 1, Floor[(
n - 1)/2]}], {n, 0, 30}]
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 *)
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 *)
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 1, 2, 3, 4, 7, 7}, 60] (* Harvey P. Dale, Jul 29 2018 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Dec 04 2010
EXTENSIONS
More terms from Colin Barker, Oct 31 2016
STATUS
approved