A140439 Array, by antidiagonals, arises in counting <= k facets in d-dimensional n-point sets.

3, 4, 9, 5, 16, 18, 6, 25, 40, 30, 7, 36, 75, 80, 45, 8, 49, 126, 175, 140, 63, 9, 64, 196, 336, 350, 224, 84, 10, 81, 288, 588, 756, 630, 336, 108, 11, 100, 405, 960, 1470, 1512, 1050, 480, 135

Offset

1,1

Comments

The formula in the cited paper, p. 10, is actually given as (d+1)*C((k+d),d) with the restriction k < floor(n/(d+1)), but that restriction is obviated in the conclusion as an artifact of the proof. The formula is a lower bound for counting <= k facets in d-dimensional n-point sets in R^d. In the table shown, the first column is (d+1) and the second column is (d+1)^2.

Links

Table of n, a(n) for n=1..45.

Oswin Aichholzer, Jesús García, David Orden and Pedro Ramos, New results on lower bounds for the number of (at most k)-facets arXiv:0801.1036

Formula

A[d,n] = (d+1)*C((k+n),n).

Example

Table begins:
===============================================================================
...|.n=0.|.n=1.|.n=2.|..n=3.|..n=4.|...n=5.|...n=6.|...n=7.|....n=8.|....n=9.|.in.OEIS
===============================================================================
k=2.|...3.|...9.|..18.|...30.|...45.|....63.|....84.|...108.|....135.|....165.|A045943
k=3.|...4.|..16.|..40.|...80.|..140.|...224.|...336.|...480.|....660.|....880.|4*A000292
k=4.|...5.|..25.|..75.|..175.|..350.|...630.|..1050.|..1650.|...2475.|...3575.|
k=5.|...6.|..36.|.126.|..336.|..756.|..1512.|..2772.|..4752.|...7722.|..12012.|
k=6.|...7.|..49.|.196.|..588.|.1470.|..3234.|..6468.|.12012.|..21021.|..35035.|
k=7.|...8.|..64.|.288.|..960.|.2640.|..6336.|.13728.|.27456.|..51480.|..91520.|
k=8.|...9.|..81.|.405.|.1485.|.4455.|.11583.|.27027.|.57915.|.115830.|.218790.|
===============================================================================

Crossrefs

Cf. A000292, A045943.

Sequence in context: A183211 A256469 A192334 * A023183 A102320 A227717

Adjacent sequences: A140436 A140437 A140438 * A140440 A140441 A140442

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Jun 20 2008

Status

approved