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Number of n-element subsets of [n+5] having an even sum.
2

%I #10 Sep 30 2019 11:34:33

%S 1,3,9,28,66,126,226,396,651,1001,1491,2184,3108,4284,5796,7752,10197,

%T 13167,16797,21252,26598,32890,40326,49140,59423,71253,84903,100688,

%U 118728,139128,162248,188496,218025,250971,287793,329004,374794,425334,481194,543004

%N Number of n-element subsets of [n+5] having an even sum.

%H Alois P. Heinz, <a href="/A282081/b282081.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6,-18,38,-63,84,-92,84,-63,38,-18,6,-1).

%F G.f.: (x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^6).

%F a(n) = A282011(n+5,n).

%F a(n) = (1+n)*(2+n)*(3+n)*(4+n)*(5+n)/240 + ((-i)^n+i^n)*(8+6*n+n^2)/32 where i=sqrt(-1). - _Colin Barker_, Feb 06 2017

%e a(0) = 1: {}.

%e a(1) = 3: {2}, {4}, {6}.

%e a(2) = 9: {1,3}, {1,5}, {1,7}, {2,4}, {2,6}, {3,5}, {3,7}, {4,6}, {5,7}.

%t LinearRecurrence[{6,-18,38,-63,84,-92,84,-63,38,-18,6,-1},{1,3,9,28,66,126,226,396,651,1001,1491,2184},40] (* _Harvey P. Dale_, Sep 30 2019 *)

%o (PARI) Vec((x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1) / ((x^2+1)^3*(x-1)^6) + O(x^60)) \\ _Colin Barker_, Feb 06 2017

%Y Cf. A282011.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Feb 05 2017