|
%I #8 Oct 04 2015 23:00:44
%S -1,-1,0,-1,1,3,-1,2,5,8,-1,3,7,11,15,-1,4,9,14,19,24,-1,5,11,17,23,
%T 29,35,-1,6,13,20,27,34,41,48,-1,7,15,23,31,39,47,55,63,-1,8,17,26,35,
%U 44,53,62,71,80,-1,9,19,29,39,49,59,69,79,89,99,-1,10,21,32,43,54,65,76,87
%N Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.
%C Arises in complete intersection threefolds,
%C Also can be produced as a triangle read by rows: a(n, k) = nk - (n + k). - _Alonso del Arte_, Jul 09 2009
%C Kosta: Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively. with n=>k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)*(n-1) - 1 singular points, then it is factorial.
%H Dimitra Kosta, <a href="http://arxiv.org/abs/0808.4071">Factoriality of complete intersection threefolds</a> arXiv:0808.4071 [math.AG]
%F A[k,n] = (n+k-2)*(n-1) - 1.
%F Antidiagonal sum: Sum_{n=1..d} A(d-n+1,n) = d*(d^2-2d-1)/2 = -A110427(d). - _R. J. Mathar_, Jul 10 2009
%e From _R. J. Mathar_, Jul 10 2009: (Start)
%e The rows A(n,1), A(n,2), A(n,3), etc., are :
%e .-1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
%e .-1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
%e .-1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
%e .-1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
%e .-1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
%e .-1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
%e .-1...6..15..26..39..54..71..90.111.134.159.186.215.246
%e .-1...7..17..29..43..59..77..97.119.143.169.197.227.259
%e .-1...8..19..32..47..64..83.104.127.152.179.208.239.272
%e .-1...9..21..35..51..69..89.111.135.161.189.219.251.285
%e .-1..10..23..38..55..74..95.118.143.170.199.230.263.298
%e .-1..11..25..41..59..79.101.125.151.179.209.241.275.311
%e .-1..12..27..44..63..84.107.132.159.188.219.252.287.324
%e .-1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
%e (End)
%e As a triangle:
%e . 0
%e . 1, 3
%e . 2, 5, 8
%e . 3, 7, 11, 15
%e . 4, 9, 14, 19, 24
%e . 5, 11, 17, 23, 29, 35
%e . 6, 13, 20, 27, 34, 41, 48
%e . 7, 15, 23, 31, 39, 47, 55, 63
%e . 8, 17, 26, 35, 44, 53, 62, 71, 80
%p A := proc(k,n) (n+k-2)*(n-1)-1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d,",A(d-n+1,n)) ; od: od: # _R. J. Mathar_, Jul 10 2009
%t a[n_, k_] := a[n, k] = n*k - (n + k); ColumnForm[Table[a[n, k], {n, 10}, {k, n}], Center] (* _Alonso del Arte_, Jul 09 2009 *)
%Y Row 1 = A067998(n) for n>0. Row 2 = A028387(n) for n>0.Column 1 = -A000012(n). Column 2 = A001477. Column 3 = A005408(k). Column 4 = A016789(k+1). Column 5 = A004767(k+2). Column 6 = A016897(k+3). Column 7 = A016969(k+4). Column 8 = A017053(k+5). Column 9 = A004771(k+6). Column 10 = A017257(k+7).
%Y Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.
%K easy,sign,tabl
%O 1,6
%A _Jonathan Vos Post_, Sep 13 2008
%E Duplicate of 6th antidiagonal removed by _R. J. Mathar_, Jul 10 2009
%E Keyword:tabl added by _R. J. Mathar_, Jul 23 2009
%E Edited by _N. J. A. Sloane_, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.
|
