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
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Feb 29 2020
STATUS
approved