

A144204


Array A(k,n) = (n+k2)*(n1)  1 (k >= 1, n >= 1) read by antidiagonals.


1



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, 29, 35, 1, 6, 13, 20, 27, 34, 41, 48, 1, 7, 15, 23, 31, 39, 47, 55, 63, 1, 8, 17, 26, 35, 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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,6


COMMENTS

Arises in complete intersection threefolds,
Also can be produced as a triangle read by rows: a(n, k) = nk  (n + k).  Alonso del Arte, Jul 09 2009
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+k2)*(n1)  1 singular points, then it is factorial.


LINKS



FORMULA

A[k,n] = (n+k2)*(n1)  1.
Antidiagonal sum: Sum_{n=1..d} A(dn+1,n) = d*(d^22d1)/2 = A110427(d).  R. J. Mathar, Jul 10 2009


EXAMPLE

The rows A(n,1), A(n,2), A(n,3), etc., are :
.1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
.1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
.1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
.1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
.1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
.1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
.1...6..15..26..39..54..71..90.111.134.159.186.215.246
.1...7..17..29..43..59..77..97.119.143.169.197.227.259
.1...8..19..32..47..64..83.104.127.152.179.208.239.272
.1...9..21..35..51..69..89.111.135.161.189.219.251.285
.1..10..23..38..55..74..95.118.143.170.199.230.263.298
.1..11..25..41..59..79.101.125.151.179.209.241.275.311
.1..12..27..44..63..84.107.132.159.188.219.252.287.324
.1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
(End)
As a triangle:
. 0
. 1, 3
. 2, 5, 8
. 3, 7, 11, 15
. 4, 9, 14, 19, 24
. 5, 11, 17, 23, 29, 35
. 6, 13, 20, 27, 34, 41, 48
. 7, 15, 23, 31, 39, 47, 55, 63
. 8, 17, 26, 35, 44, 53, 62, 71, 80


MAPLE

A := proc(k, n) (n+k2)*(n1)1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d, ", A(dn+1, n)) ; od: od: # R. J. Mathar, Jul 10 2009


MATHEMATICA

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 *)


CROSSREFS

Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.


KEYWORD



AUTHOR



EXTENSIONS

Duplicate of 6th antidiagonal removed by R. J. Mathar, Jul 10 2009
Edited by N. J. A. Sloane, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.


STATUS

approved



