OFFSET
1,4
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) = binomial(n - 1, n/2) when n is even, a(n) = binomial(n - 1, (n + 1)/2) when n is 3 mod 4, and a(n) = binomial(n - 1, (n - 1)/2) when n is 1 mod 4.
A214282(n) - A214283(n) is A056040(n) if n is even and A056040(n)/((n+1)/2) otherwise. - Peter Luschny, Jul 08 2016
EXAMPLE
G.f. = x + x^2 + x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 35*x^8 + ...
MATHEMATICA
Table[{Binomial[n - 1, n/2], Binomial[n, n/2], Binomial[n + 1, n/2 + 1], Binomial[n + 2, n/2 + 2]}, {n, 0, 28, 4}] (* Alonso del Arte, Jul 09 2012 *)
PROG
(PARI) a(n)=binomial(n-1, if(n%2, (n+1)\4*2, n/2)) \\ Charles R Greathouse IV, Jul 09 2012
(PARI) {a(n) = if( n<1, 0, vecmax( Vec((1 - x)^(n-1))))}; /* Michael Somos, Apr 21 2014 */
(Haskell)
a214282 n = a007318 (n - 1) (a004524 (n - 1))
-- Reinhard Zumkeller, Jul 14 2012
(Python)
from math import comb
def A214282(n): return comb(n-1, (n+1>>1)&(-1^(n&1))) # Chai Wah Wu, Jan 31 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Terence Tao, Jul 09 2012
STATUS
approved