A333319 a(n) is the number of subsets of {1..n} that contain exactly 3 odd and 1 even numbers.

0, 0, 0, 0, 0, 2, 3, 12, 16, 40, 50, 100, 120, 210, 245, 392, 448, 672, 756, 1080, 1200, 1650, 1815, 2420, 2640, 3432, 3718, 4732, 5096, 6370, 6825, 8400, 8960, 10880, 11560, 13872, 14688, 17442, 18411, 21660, 22800, 26600, 27930, 32340, 33880, 38962, 40733, 46552

Offset

0,6

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 (1,4,-4,-6,6,4,-4,-1,1).

Formula

a(n) = binomial(ceiling(n/2),3) * binomial(floor(n/2), 1).

From Colin Barker, Mar 15 2020: (Start)

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

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

(End)

Example

a(7)=12 and the 12 subsets are {1,2,3,5}, {1,2,3,7}, {1,2,5,7}, {1,3,4,5}, {1,3,4,7}, {1,3,5,6}, {1,3,6,7}, {1,4,5,7}, {1,5,6,7}, {2,3,5,7}, {3,4,5,7}, {3,5,6,7}.

Mathematica

Array[Binomial[Ceiling[#], 3] Binomial[Floor[#], 1] &[#/2] &, 48, 0] (* Michael De Vlieger, Mar 14 2020 *)

Prog

(PARI) concat([0, 0, 0, 0, 0], Vec(x^5*(2 + x + x^2) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Mar 15 2020

Crossrefs

Cf. A330299.

Sequence in context: A123761 A181121 A345362 * A047163 A046486 A073452

Adjacent sequences: A333316 A333317 A333318 * A333320 A333321 A333322

Keyword

nonn,easy

Author

Enrique Navarrete, Mar 14 2020

Status

approved