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A333320 a(n) is the number of subsets of {1..n} that contain exactly 4 odd and 1 even numbers.

%I A333320 #17 May 01 2020 22:58:18
%S A333320 0,0,0,0,0,0,0,3,4,20,25,75,90,210,245,490,560,1008,1134,1890,2100,
%T A333320 3300,3630,5445,5940,8580,9295,13013,14014,19110,20475,27300,29120,
%U A333320 38080,40460,52020,55080,69768,73644,92055,96900,119700,125685,153615,160930,194810,203665
%N A333320 a(n) is the number of subsets of {1..n} that contain exactly 4 odd and 1 even numbers.
%C A333320 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 A333320 Colin Barker, <a href="/A333320/b333320.txt">Table of n, a(n) for n = 0..1000</a>
%H A333320 <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 A333320 a(n) = binomial(ceiling(n/2),4) * floor(n/2).
%F A333320 From _Colin Barker_, Mar 17 2020: (Start)
%F A333320 G.f.: x^7*(3 + x + x^2) / ((1 - x)^6*(1 + x)^5).
%F A333320 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 A333320 (End)
%e A333320 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 A333320 Array[Binomial[Ceiling[#], 4] Binomial[Floor[#], 1] &[#/2] &, 47, 0] (* _Michael De Vlieger_, Mar 14 2020 *)
%o A333320 (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 A333320 Cf. A333321.
%K A333320 nonn,easy
%O A333320 0,8
%A A333320 _Enrique Navarrete_, Mar 14 2020