cp's OEIS Frontend

A159914 Half the number of (n-3)-element subsets of {1,...,n} whose elements sum up to an odd value.

%I A159914 #20 May 10 2020 10:16:15
%S A159914 0,0,0,0,1,3,5,8,14,22,30,40,55,73,91,112,140,172,204,240,285,335,385,
%T A159914 440,506,578,650,728,819,917,1015,1120,1240,1368,1496,1632,1785,1947,
%U A159914 2109,2280,2470,2670,2870,3080,3311,3553,3795,4048,4324,4612,4900,5200
%N A159914 Half the number of (n-3)-element subsets of {1,...,n} whose elements sum up to an odd value.
%C A159914 Half the preantepenultimate column, i.e., T(n, n-3), of the triangle defined in A159916.
%H A159914 Simon Plouffe, <a href="http://vixra.org/abs/1409.0048"> Conjectures of the OEIS, as of June 20, 2018.</a>
%H A159914 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-8,12,-14,12,-8,4,-1).
%F A159914 G.f.: x^4*(1-x+x^2)/((1-x)^4*(1+x^2)^2).
%F A159914 a(n) = A159916(n(n-1)/2+n-3)/2 = T(n,n-3)/2 as defined there.
%F A159914 a(2k) = k(k-1)(2k-1)/6.
%F A159914 Euler transform of 3 - x + x^2 + 2*x^3 - x^5. - _Simon Plouffe_, Jun 22 2018
%e A159914 The first nontrivial term a(4)=1 is half the number of 4-3=1-element subsets of {1,2,3,4} whose elements have an odd sum: {1} and {3}.
%e A159914 a(5)=3 is half the number of 5-3=2-element subsets of {1,2,3,4,5} whose elements have an odd sum: {1,2}, {1,4}, {2,3}, {2,5}, {3,4} and {4,5}.
%o A159914 (PARI) A159914(n)=polcoeff((1-x+x^2)/(1-x)^4/(1+x^2)^2+O(x^(n-3)),n-4)
%Y A159914 Cf. A228705 (counts subsets with even sum).
%K A159914 nonn
%O A159914 0,6
%A A159914 _M. F. Hasler_, May 02 2009