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A330300 a(n) is the number of subsets of {1..n} that contain exactly 2 odd and 3 even numbers.

%I A330300 #34 Mar 15 2020 10:53:02
%S A330300 0,0,0,0,0,0,3,6,24,40,100,150,300,420,735,980,1568,2016,3024,3780,
%T A330300 5400,6600,9075,10890,14520,17160,22308,26026,33124,38220,47775,54600,
%U A330300 67200,76160,92480,104040,124848,139536,165699,184110,216600,239400,279300,307230,355740,389620,448063
%N A330300 a(n) is the number of subsets of {1..n} that contain exactly 2 odd and 3 even numbers.
%C A330300 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 A330300 Colin Barker, <a href="/A330300/b330300.txt">Table of n, a(n) for n = 0..1000</a>
%H A330300 <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 A330300 a(n) = binomial(ceiling(n/2), 2) * binomial(floor(n/2), 3).
%F A330300 From _Colin Barker_, Mar 01 2020: (Start)
%F A330300 G.f.: x^6*(3 + 3*x + 3*x^2 + x^3) / ((1 - x)^6*(1 + x)^5).
%F A330300 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 A330300 (End)
%F A330300 E.g.f.: (x*(-15 + 3*x - 7*x^2 + 2*x^3 + x^4)*cosh(x) + (15 - 3*x + 12*x^2 - 3*x^3 + x^4 + x^5)*sinh(x))/384. - _Stefano Spezia_, Mar 02 2020
%e A330300 a(7) = 6 and the 6 subsets are {1,2,3,4,6}, {1,2,4,5,6}, {1,2,4,6,7}, {2,3,4,5,6}, {2,3,4,6,7}, {2,4,5,6,7}.
%t A330300 a[n_] := Binomial[Ceiling[n/2], 2] * Binomial[Floor[n/2], 3]; Array[a, 47, 0] (* _Amiram Eldar_, Mar 01 2020 *)
%o A330300 (PARI) a(n) = binomial(ceil(n/2), 2) * binomial(floor(n/2), 3) \\ _Andrew Howroyd_, Mar 01 2020
%o A330300 (PARI) concat([0,0,0,0,0,0], Vec(x^6*(3 + 3*x + 3*x^2 + x^3) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ _Colin Barker_, Mar 02 2020
%Y A330300 Cf. A330298, A330299.
%K A330300 nonn,easy
%O A330300 0,7
%A A330300 _Enrique Navarrete_, Feb 29 2020