A333321 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 4 even numbers.

0, 0, 0, 0, 0, 0, 0, 0, 4, 5, 25, 30, 90, 105, 245, 280, 560, 630, 1134, 1260, 2100, 2310, 3630, 3960, 5940, 6435, 9295, 10010, 14014, 15015, 20475, 21840, 29120, 30940, 40460, 42840, 55080, 58140, 73644, 77520, 96900, 101745, 125685, 131670, 160930, 168245, 203665, 212520

Offset

0,9

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

From Colin Barker, Mar 17 2020: (Start)

G.f.: x^8*(4 + x) / ((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(9)=5 and the 5 subsets are {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {2,4,6,8,9}.

Mathematica

Array[Binomial[Ceiling[#], 1] Binomial[Floor[#], 4] &[#/2] &, 48, 0] (* Michael De Vlieger, Mar 14 2020 *)

Prog

(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(x^8*(4 + x) / ((1 - x)^6*(1 + x)^5) + O(x^45))) \\ Colin Barker, Mar 17 2020

Crossrefs

Cf. A330298, A330299, A333320.

Sequence in context: A219515 A248246 A164054 * A047168 A220538 A187996

Adjacent sequences: A333318 A333319 A333320 * A333322 A333323 A333324

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 14 2020

Status

approved