A330299 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 3 even numbers.

0, 0, 0, 0, 0, 0, 3, 4, 16, 20, 50, 60, 120, 140, 245, 280, 448, 504, 756, 840, 1200, 1320, 1815, 1980, 2640, 2860, 3718, 4004, 5096, 5460, 6825, 7280, 8960, 9520, 11560, 12240, 14688, 15504, 18411, 19380, 22800, 23940, 27930, 29260, 33880, 35420, 40733, 42504, 48576, 50600, 57500

Offset

0,7

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

From Colin Barker, Mar 01 2020: (Start)

G.f.: x^6*(3 + x) / ((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)

E.g.f.: (x*(15 - 3*x - 2*x^2 + x^3)*cosh(x) + (-15 + 3*x - 3*x^2 + x^4)*sinh(x))/96. - Stefano Spezia, Mar 02 2020

Example

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

Mathematica

a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 3]; Array[a, 51, 0] (* Amiram Eldar, Mar 01 2020 *)

Prog

(PARI) a(n) = ceil(n/2) * binomial(floor(n/2), 3) \\ Andrew Howroyd, Mar 01 2020
(PARI) concat([0, 0, 0, 0, 0, 0], Vec(x^6*(3 + x) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Mar 02 2020

Crossrefs

Cf. A330298, A330300.

Sequence in context: A325212 A329890 A334319 * A047164 A304988 A108653

Adjacent sequences: A330296 A330297 A330298 * A330300 A330301 A330302

Keyword

nonn,easy

Author

Enrique Navarrete, Feb 29 2020

Status

approved