%I #32 Mar 15 2020 11:57:37
%S 0,0,0,0,0,0,3,4,16,20,50,60,120,140,245,280,448,504,756,840,1200,
%T 1320,1815,1980,2640,2860,3718,4004,5096,5460,6825,7280,8960,9520,
%U 11560,12240,14688,15504,18411,19380,22800,23940,27930,29260,33880,35420,40733,42504,48576,50600,57500
%N a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 3 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="/A330299/b330299.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) = ceiling(n/2) * binomial(floor(n/2), 3).
%F From _Colin Barker_, Mar 01 2020: (Start)
%F G.f.: x^6*(3 + x) / ((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)
%F E.g.f.: (x*(15 - 3*x - 2*x^2 + x^3)*cosh(x) + (-15 + 3*x - 3*x^2 + x^4)*sinh(x))/96. - _Stefano Spezia_, Mar 02 2020
%e a(7)=4 and the 4 subsets are {1,2,4,6}, {2,3,4,6}, {2,4,5,6}, {2,4,6,7}.
%t a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 3]; Array[a, 51, 0] (* _Amiram Eldar_, Mar 01 2020 *)
%o (PARI) a(n) = ceil(n/2) * binomial(floor(n/2), 3) \\ _Andrew Howroyd_, Mar 01 2020
%o (PARI) concat([0,0,0,0,0,0], Vec(x^6*(3 + x) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Mar 02 2020
%Y Cf. A330298, A330300.
%K nonn,easy
%O 0,7
%A _Enrique Navarrete_, Feb 29 2020