cp's OEIS Frontend

A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

%I A214283 #33 Jan 31 2024 15:55:03
%S A214283 0,-1,-2,-3,-4,-10,-20,-35,-56,-126,-252,-462,-792,-1716,-3432,-6435,
%T A214283 -11440,-24310,-48620,-92378,-167960,-352716,-705432,-1352078,
%U A214283 -2496144,-5200300,-10400600,-20058300,-37442160,-77558760,-155117520,-300540195
%N A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.
%C A214283 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.
%H A214283 Reinhard Zumkeller, <a href="/A214283/b214283.txt">Table of n, a(n) for n = 1..1000</a>
%H A214283 Terry Tao, <a href="http://mathoverflow.net/questions/101787/optimal-bounds-for-an-alternating-sum-on-a-downset">Optimal bounds for an alternating sum on a downset</a>, 2012.
%F A214283 a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),
%F A214283 a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),
%F A214283 a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).
%F A214283 a(n) = A214282(n) - A001405(n). - _Reinhard Zumkeller_, Jul 14 2012
%F A214283 For n > 1: a(n) = - A007318(n-1, A004525(n)). - _Reinhard Zumkeller_, Jul 14 2012
%F A214283 a(n+1) = -A000108(n/2) * A212831(n). - _Paul Curtz_, Nov 04 2012
%t A214283 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_ *)
%o A214283 (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
%o A214283 (Haskell)
%o A214283 a214283 1 = 0
%o A214283 a214283 n = - a007318 (n - 1) (a004525 n)
%o A214283 -- _Reinhard Zumkeller_, Jul 14 2012
%o A214283 (Python)
%o A214283 from math import comb
%o A214283 def A214283(n): return -comb(n-1,(n>>1)|(n&1)) # _Chai Wah Wu_, Jan 31 2024
%Y A214283 Cf. A000108, A001405, A004525, A007318, A212831, A214282.
%K A214283 sign
%O A214283 1,3
%A A214283 _Terence Tao_, Jul 09 2012