A330298 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 2 even numbers.

0, 0, 0, 0, 2, 3, 9, 12, 24, 30, 50, 60, 90, 105, 147, 168, 224, 252, 324, 360, 450, 495, 605, 660, 792, 858, 1014, 1092, 1274, 1365, 1575, 1680, 1920, 2040, 2312, 2448, 2754, 2907, 3249, 3420, 3800, 3990, 4410, 4620, 5082, 5313, 5819, 6072, 6624, 6900, 7500, 7800, 8450, 8775, 9477

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

			For n=6, a(6) = 9 and the 9 subsets are: {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• J.S. Seneschal, Oblong Prism Illustration
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A330299, A330300.

Cf. A000217, A002378.

Interleaves the positive integers of A006002 and A027480.

Programs

Mathematica

a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 2]; Array[a, 55, 0] (* Amiram Eldar, Mar 01 2020 *)
Table[Length[Select[Subsets[Range[n],{3}],Total[Boole[OddQ[#]]]==1&]],{n,0,60}] (* Harvey P. Dale, Jul 26 2020 *)

PARI

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

PARI

concat([0,0,0,0], Vec(x^4*(2 + x) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Mar 02 2020

Formula

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

From Colin Barker, Mar 01 2020: (Start)

G.f.: x^4*(2 + x) / ((1 - x)^4*(1 + x)^3).

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

E.g.f.: (x*(-3 + x + x^2)*cosh(x) + (3 - x + x^3)*sinh(x))/16. - Stefano Spezia, Mar 02 2020

a(n) = 3/32-5*n^2/32-n/32+n^3/16+(-)^n*(n-3+n^2)/32. - R. J. Mathar, Mar 31 2025