A333320 - OEIS

%I #17 May 01 2020 22:58:18

%S 0,0,0,0,0,0,0,3,4,20,25,75,90,210,245,490,560,1008,1134,1890,2100,

%T 3300,3630,5445,5940,8580,9295,13013,14014,19110,20475,27300,29120,

%U 38080,40460,52020,55080,69768,73644,92055,96900,119700,125685,153615,160930,194810,203665

%N a(n) is the number of subsets of {1..n} that contain exactly 4 odd and 1 even numbers.

%C The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

%H Colin Barker, <a href="/A333320/b333320.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) = binomial(ceiling(n/2),4) * floor(n/2).

%F From _Colin Barker_, Mar 17 2020: (Start)

%F G.f.: x^7*(3 + x + x^2) / ((1 - x)^6*(1 + x)^5).

%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) for n>10.

%F (End)

%e a(8)=4 and the 4 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,6,7}, {1,3,5,7,8}.

%t Array[Binomial[Ceiling[#], 4] Binomial[Floor[#], 1] &[#/2] &, 47, 0] (* _Michael De Vlieger_, Mar 14 2020 *)

%o (PARI) concat([0,0,0,0,0,0,0], Vec(x^7*(3 + x + x^2) / ((1 - x)^6*(1 + x)^5) + O(x^50))) \\ _Colin Barker_, Mar 17 2020

%Y Cf. A333321.

%K nonn,easy

%O 0,8

%A _Enrique Navarrete_, Mar 14 2020