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

%I #20 May 10 2020 10:16:15

%S 0,0,0,0,1,3,5,8,14,22,30,40,55,73,91,112,140,172,204,240,285,335,385,

%T 440,506,578,650,728,819,917,1015,1120,1240,1368,1496,1632,1785,1947,

%U 2109,2280,2470,2670,2870,3080,3311,3553,3795,4048,4324,4612,4900,5200

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

%C Half the preantepenultimate column, i.e., T(n, n-3), of the triangle defined in A159916.

%H Simon Plouffe, <a href="http://vixra.org/abs/1409.0048"> Conjectures of the OEIS, as of June 20, 2018.</a>

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

%F a(n) = A159916(n(n-1)/2+n-3)/2 = T(n,n-3)/2 as defined there.

%F a(2k) = k(k-1)(2k-1)/6.

%F Euler transform of 3 - x + x^2 + 2*x^3 - x^5. - _Simon Plouffe_, Jun 22 2018

%e 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 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 (PARI) A159914(n)=polcoeff((1-x+x^2)/(1-x)^4/(1+x^2)^2+O(x^(n-3)),n-4)

%Y Cf. A228705 (counts subsets with even sum).

%K nonn

%O 0,6

%A _M. F. Hasler_, May 02 2009