A028725 a(n) = floor(n/2) * floor((n-1)/2) * floor((n-2)/2) * floor((n-3)/2) * floor((n-4)/2) / 12.

0, 0, 0, 0, 0, 0, 1, 3, 12, 24, 60, 100, 200, 300, 525, 735, 1176, 1568, 2352, 3024, 4320, 5400, 7425, 9075, 12100, 14520, 18876, 22308, 28392, 33124, 41405, 47775, 58800, 67200, 81600, 92480, 110976, 124848, 148257, 165699, 194940, 216600, 252700, 279300

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Crossrefs

Bisections: A004282, A004302.

Cf. A002620, A028724.

Programs

Magma

[(&*[Floor((n-j)/2):j in [0..4]])/12: n in [0..60]]; // G. C. Greubel, Apr 08 2022

Mathematica

Table[(Times@@Floor/@(n/2-Range[0,4]/2))/12,{n,0,50}] (* or *) LinearRecurrence[ {1,5,-5,-10,10,10,-10,-5,5,1,-1}, {0,0,0,0,0,0,1,3,12,24,60}, 50] (* Harvey P. Dale, Jun 26 2012 *)

PARI

concat([0,0,0,0,0,0], Vec(x^6*(x^4+2*x^3+4*x^2+2*x+1)/((x-1)^6*(x+1)^5) + O(x^100))) \\ Colin Barker, Mar 01 2015

SageMath

[(1/768)*((-1)^n*(45 -65*n +38*n^2 -10*n^3 +n^4) -45 +193*n -230*n^2 +114*n^3 -25*n^4 +2*n^5) for n in (0..60)] # G. C. Greubel, Apr 08 2022

Formula

a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=1, a(7)=3, a(8)=12, a(9)=24, a(10)=60. - Harvey P. Dale, Jun 26 2012

G.f.: x^6*(1+2*x+4*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^5). - Colin Barker, Mar 01 2015

From R. J. Mathar, Sep 23 2021: (Start)

a(2*n+1) = A004282(n-2).

a(2*n) = A004302(n-2).

a(n) = A028724(n)*A002620(n-4)/6. (End)

From G. C. Greubel, Apr 08 2022: (Start)

a(n) = (1/768)*((-1)^n*(45 -65*n +38*n^2 -10*n^3 +n^4) -45 +193*n -230*n^2 +114*n^3 -25*n^4 +2*n^5).

E.g.f.: (1/768)*((45 +36*x +15*x^2 +4*x^3 +x^4)*exp(-x) + (-45 +54*x -33*x^2 + 14*x^3 -5*x^4 +2*x^5)*exp(x)). (End)