A282080 - OEIS

%I #11 Jan 01 2024 20:05:34

%S 1,2,6,19,38,60,100,170,255,350,490,693,924,1176,1512,1956,2445,2970,

%T 3630,4455,5346,6292,7436,8814,10283,11830,13650,15785,18040,20400,

%U 23120,26248,29529,32946,36822,41211,45790,50540,55860,61810,67991,74382,81466,89309

%N Number of n-element subsets of [n+4] having an even sum.

%H Alois P. Heinz, <a href="/A282080/b282080.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (5,-13,25,-38,46,-46,38,-25,13,-5,1).

%F G.f.: -(x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^5).

%F a(n) = A282011(n+4,n).

%F a(n) = (2*(1+n)*(2+n)*(3+n)*(4+n) + 3*(-i*(-i)^n*((3+8*i) + (4+6*i)*n + (1+i)*n^2) + i^n*((8+3*i) + (6+4*i)*n + (1+i)*n^2)))/96 where i=sqrt(-1). - _Colin Barker_, Feb 06 2017

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

%e a(3) = 19: {1,2,3}, {1,2,5}, {1,2,7}, {1,3,4}, {1,3,6}, {1,4,5}, {1,4,7}, {1,5,6}, {1,6,7}, {2,3,5}, {2,3,7}, {2,4,6}, {2,5,7}, {3,4,5}, {3,4,7}, {3,5,6}, {3,6,7}, {4,5,7}, {5,6,7}.

%t CoefficientList[Series[-(x^2 - x + 1)*(x^4 - 2*x^3 + 6*x^2 - 2*x + 1)/((x^2 + 1)^3*(x - 1)^5), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jan 01 2024 *)

%o (PARI) Vec(-(x^2-x+1)*(x^4-2*x^3+6*x^2-2*x+1)/((x^2+1)^3*(x-1)^5) + O(x^90)) \\ _Colin Barker_, Feb 06 2017

%Y Cf. A282011.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Feb 05 2017