OFFSET
1,3
COMMENTS
An m-downset is a set of subsets of 1..m such that if S is in the set, so are all subsets of S. The Euler characteristic of a downset is the number of sets in the downset with an even cardinality, minus the number with an odd cardinality.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Terry Tao, Optimal bounds for an alternating sum on a downset, 2012.
FORMULA
a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),
a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),
a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).
MATHEMATICA
A212831[n_] := (1/4)*((-(1+(-1)^n))*(-1+(-1)^Floor[n/2]) - (-3+(-1)^n)*n); a[n_] := -CatalanNumber[Floor[(n-1)/2]]*A212831[n-1]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Nov 06 2012, after Paul Curtz *)
PROG
(PARI) a(n)=-binomial(n-1, if(n%2, if(n%4==3, n-1, n+1), n)/2) \\ Charles R Greathouse IV, Jul 10 2012
(Haskell)
a214283 1 = 0
a214283 n = - a007318 (n - 1) (a004525 n)
-- Reinhard Zumkeller, Jul 14 2012
(Python)
from math import comb
def A214283(n): return -comb(n-1, (n>>1)|(n&1)) # Chai Wah Wu, Jan 31 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Terence Tao, Jul 09 2012
STATUS
approved