A028725 - OEIS

%I #19 Apr 09 2022 01:27:25

%S 0,0,0,0,0,0,1,3,12,24,60,100,200,300,525,735,1176,1568,2352,3024,

%T 4320,5400,7425,9075,12100,14520,18876,22308,28392,33124,41405,47775,

%U 58800,67200,81600,92480,110976,124848,148257,165699,194940,216600,252700,279300

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

%H Harvey P. Dale, <a href="/A028725/b028725.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

%F 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

%F 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

%F From _R. J. Mathar_, Sep 23 2021: (Start)

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

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

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

%F From _G. C. Greubel_, Apr 08 2022: (Start)

%F 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).

%F 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)

%t 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 *)

%o (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

%o (Magma) [(&*[Floor((n-j)/2):j in [0..4]])/12: n in [0..60]]; // _G. C. Greubel_, Apr 08 2022

%o (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

%Y Bisections: A004282, A004302.

%Y Cf. A002620, A028724.

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_