A330300 a(n) is the number of subsets of {1..n} that contain exactly 2 odd and 3 even numbers.

0, 0, 0, 0, 0, 0, 3, 6, 24, 40, 100, 150, 300, 420, 735, 980, 1568, 2016, 3024, 3780, 5400, 6600, 9075, 10890, 14520, 17160, 22308, 26026, 33124, 38220, 47775, 54600, 67200, 76160, 92480, 104040, 124848, 139536, 165699, 184110, 216600, 239400, 279300, 307230, 355740, 389620, 448063

Offset: 0

Author

Enrique Navarrete, Feb 29 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(7) = 6 and the 6 subsets are {1,2,3,4,6}, {1,2,4,5,6}, {1,2,4,6,7}, {2,3,4,5,6}, {2,3,4,6,7}, {2,4,5,6,7}.
		

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. A330298, A330299.

Programs

Mathematica

a[n_] := Binomial[Ceiling[n/2], 2] * Binomial[Floor[n/2], 3]; Array[a, 47, 0] (* Amiram Eldar, Mar 01 2020 *)

PARI

a(n) = binomial(ceil(n/2), 2) * binomial(floor(n/2), 3) \\ Andrew Howroyd, Mar 01 2020

PARI

concat([0,0,0,0,0,0], Vec(x^6*(3 + 3*x + 3*x^2 + x^3) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ Colin Barker, Mar 02 2020

Formula

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

From Colin Barker, Mar 01 2020: (Start)

G.f.: x^6*(3 + 3*x + 3*x^2 + x^3) / ((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)

E.g.f.: (x*(-15 + 3*x - 7*x^2 + 2*x^3 + x^4)*cosh(x) + (15 - 3*x + 12*x^2 - 3*x^3 + x^4 + x^5)*sinh(x))/384. - Stefano Spezia, Mar 02 2020