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A110145 a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).

%I A110145 #18 Feb 06 2017 16:04:41
%S A110145 0,1,4,10,32,126,472,1716,6400,24310,92504,352716,1351616,5200300,
%T A110145 20060016,77558760,300533760,1166803110,4537591960,17672631900,
%U A110145 68923172032,269128937220,1052049834576,4116715363800,16123800489472,63205303218876,247959271674352
%N A110145 a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).
%C A110145 Interleaves A002458 and A037964.
%C A110145 Number of n-element subsets of [2n] having an odd sum. - _Alois P. Heinz_, Feb 06 2017
%H A110145 Alois P. Heinz, <a href="/A110145/b110145.txt">Table of n, a(n) for n = 0..1000</a>
%F A110145 a(n) = Sum_{k=0..n} C(n, k)^2*(1-(-1)^k)/2.
%F A110145 a(n) = C(2n-1, n-1)(1-(-1)^n)/2+(C(2n, n)/2-(-1)^(n/2)*C(n, floor(n/2))/2)(1+(-1)^n)/2.
%F A110145 a(n) = (binomial(2*n, n) - binomial(n, n/2)*cos(Pi*n/2))/2 = n^2 * hypergeom([1/2-n/2, 1/2-n/2, 1-n/2, 1-n/2], [1, 3/2, 3/2], 1). - _Vladimir Reshetnikov_, Oct 04 2016
%F A110145 a(n) = A159916(2n,n). - _Alois P. Heinz_, Feb 06 2017
%t A110145 Table[Sum[Binomial[n,k]^2 Mod[k,2],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Feb 21 2013 *)
%t A110145 Table[(Binomial[2 n, n] - Binomial[n, n/2] Cos[Pi n/2])/2, {n, 0, 30}] (* _Vladimir Reshetnikov_, Oct 04 2016 *)
%o A110145 (PARI) a(n) = sum(k=0, n, binomial(n, k)^2*(k % 2)); \\ _Michel Marcus_, Oct 05 2016
%Y A110145 Cf. A159916.
%K A110145 easy,nonn
%O A110145 0,3
%A A110145 _Paul Barry_, Jul 13 2005