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

0, 1, 1, 2, 3, 4, 7, 7, 14, 11, 25, 16, 41, 22, 63, 29, 92, 37, 129, 46, 175, 56, 231, 67, 298, 79, 377, 92, 469, 106, 575, 121, 696, 137, 833, 154, 987, 172, 1159, 191, 1350, 211, 1561, 232, 1793, 254, 2047, 277, 2324, 301, 2625, 326, 2951, 352, 3303, 379

Offset: 0

Author

Roger L. Bagula, Dec 04 2010

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A175685, A057979, A152271.

Programs

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 *)

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

Formula

a(n) = A057979(n+1) + binomial(n-j,j-1) with j = A004526(n-1), n>0.

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

Extensions

More terms from Colin Barker, Oct 31 2016