A333320 a(n) is the number of subsets of {1..n} that contain exactly 4 odd and 1 even numbers.

0, 0, 0, 0, 0, 0, 0, 3, 4, 20, 25, 75, 90, 210, 245, 490, 560, 1008, 1134, 1890, 2100, 3300, 3630, 5445, 5940, 8580, 9295, 13013, 14014, 19110, 20475, 27300, 29120, 38080, 40460, 52020, 55080, 69768, 73644, 92055, 96900, 119700, 125685, 153615, 160930, 194810, 203665

Offset: 0

Author

Enrique Navarrete, Mar 14 2020

Keywords

• nonn
• easy

Comments

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

Examples

			a(8)=4 and the 4 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,6,7}, {1,3,5,7,8}.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Crossrefs

Cf. A333321.

Programs

Mathematica

Array[Binomial[Ceiling[#], 4] Binomial[Floor[#], 1] &[#/2] &, 47, 0] (* Michael De Vlieger, Mar 14 2020 *)

PARI

concat([0,0,0,0,0,0,0], Vec(x^7*(3 + x + x^2) / ((1 - x)^6*(1 + x)^5) + O(x^50))) \\ Colin Barker, Mar 17 2020

Formula

a(n) = binomial(ceiling(n/2),4) * floor(n/2).

From Colin Barker, Mar 17 2020: (Start)

G.f.: x^7*(3 + x + x^2) / ((1 - x)^6*(1 + x)^5).

a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>10.

(End)