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A333321 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 4 even numbers.

%I A333321 #18 May 01 2020 16:22:02
%S A333321 0,0,0,0,0,0,0,0,4,5,25,30,90,105,245,280,560,630,1134,1260,2100,2310,
%T A333321 3630,3960,5940,6435,9295,10010,14014,15015,20475,21840,29120,30940,
%U A333321 40460,42840,55080,58140,73644,77520,96900,101745,125685,131670,160930,168245,203665,212520
%N A333321 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 4 even numbers.
%C A333321 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 A333321 Colin Barker, <a href="/A333321/b333321.txt">Table of n, a(n) for n = 0..1000</a>
%H A333321 <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 A333321 a(n) = ceiling(n/2) * binomial(floor(n/2), 4).
%F A333321 From _Colin Barker_, Mar 17 2020: (Start)
%F A333321 G.f.: x^8*(4 + x) / ((1 - x)^6*(1 + x)^5).
%F A333321 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 A333321 (End)
%e A333321 a(9)=5 and the 5 subsets are {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {2,4,6,8,9}.
%t A333321 Array[Binomial[Ceiling[#], 1] Binomial[Floor[#], 4] &[#/2] &, 48, 0] (* _Michael De Vlieger_, Mar 14 2020 *)
%o A333321 (PARI) concat([0,0,0,0,0,0,0,0], Vec(x^8*(4 + x) / ((1 - x)^6*(1 + x)^5) + O(x^45))) \\ _Colin Barker_, Mar 17 2020
%Y A333321 Cf. A330298, A330299, A333320.
%K A333321 nonn,easy
%O A333321 0,9
%A A333321 _Enrique Navarrete_, Mar 14 2020