A282080 Number of n-element subsets of [n+4] having an even sum.

1, 2, 6, 19, 38, 60, 100, 170, 255, 350, 490, 693, 924, 1176, 1512, 1956, 2445, 2970, 3630, 4455, 5346, 6292, 7436, 8814, 10283, 11830, 13650, 15785, 18040, 20400, 23120, 26248, 29529, 32946, 36822, 41211, 45790, 50540, 55860, 61810, 67991, 74382, 81466, 89309

Offset: 0

Author

Alois P. Heinz, Feb 05 2017

Keywords

• nonn
• easy

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-13,25,-38,46,-46,38,-25,13,-5,1).

Crossrefs

Cf. A282011.

Programs

Mathematica

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 *)

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

Formula

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

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

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