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

0,8

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).

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).

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)

Example

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}.

Mathematica

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

Prog

(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

Crossrefs

Cf. A333321.

Sequence in context: A051719 A336619 A240970 * A047165 A124631 A262033

Adjacent sequences: A333317 A333318 A333319 * A333321 A333322 A333323

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 14 2020

Status

approved