A333319 - OEIS

%I #13 Apr 15 2020 08:32:17

%S 0,0,0,0,0,2,3,12,16,40,50,100,120,210,245,392,448,672,756,1080,1200,

%T 1650,1815,2420,2640,3432,3718,4732,5096,6370,6825,8400,8960,10880,

%U 11560,13872,14688,17442,18411,21660,22800,26600,27930,32340,33880,38962,40733,46552

%N a(n) is the number of subsets of {1..n} that contain exactly 3 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="/A333319/b333319.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).

%F a(n) = binomial(ceiling(n/2),3) * binomial(floor(n/2), 1).

%F From _Colin Barker_, Mar 15 2020: (Start)

%F G.f.: x^5*(2 + x + x^2) / ((1 - x)^5*(1 + x)^4).

%F a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>8.

%F (End)

%e a(7)=12 and the 12 subsets are {1,2,3,5}, {1,2,3,7}, {1,2,5,7}, {1,3,4,5}, {1,3,4,7}, {1,3,5,6}, {1,3,6,7}, {1,4,5,7}, {1,5,6,7}, {2,3,5,7}, {3,4,5,7}, {3,5,6,7}.

%t Array[Binomial[Ceiling[#], 3] Binomial[Floor[#], 1] &[#/2] &, 48, 0] (* _Michael De Vlieger_, Mar 14 2020 *)

%o (PARI) concat([0,0,0,0,0], Vec(x^5*(2 + x + x^2) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Mar 15 2020

%Y Cf. A330299.

%K nonn,easy

%O 0,6

%A _Enrique Navarrete_, Mar 14 2020