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
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Mar 14 2020
STATUS
approved