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A214282 Largest Euler characteristic of a downset on an n-dimensional cube. 9
1, 1, 1, 3, 6, 10, 15, 35, 70, 126, 210, 462, 924, 1716, 3003, 6435, 12870, 24310, 43758, 92378, 184756, 352716, 646646, 1352078, 2704156, 5200300, 9657700, 20058300, 40116600, 77558760, 145422675 (list; graph; refs; listen; history; text; internal format)
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.

a(n) = A214283(n) + A001405(n). - Reinhard Zumkeller, Jul 14 2012

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.

a(2n) = A001700(n-1). a(4n+1) = A001448(n). a(4n+3) = A186231(n).

a(n) = A007318(n-1, A004524(n-1)). - Reinhard Zumkeller, Jul 14 2012

a(n+1) = A000108([n/2])*A215495(n). - M. F. Hasler, Aug 25 2012

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

CROSSREFS

Cf. A214283.

Sequence in context: A261632 A058576 A230364 * A130200 A202269 A151375

Adjacent sequences:  A214279 A214280 A214281 * A214283 A214284 A214285

KEYWORD

nonn

AUTHOR

Terence Tao, Jul 09 2012

STATUS

approved

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Last modified October 19 07:56 EDT 2018. Contains 316337 sequences. (Running on oeis4.)