A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset

0,5

Comments

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:

.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1

.1,....3,......5,.......7,........9,.......11,........13,........15,........17

.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217

.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281

.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101

.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153

.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469

.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321

.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407

This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

Links

Table of n, a(n) for n=0..62.

R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Example

A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932

Crossrefs

Cf. A008458 (row k=2), A010006 (row k=3), A110907.

Sequence in context: A276436 A161121 A021500 * A367795 A343825 A339031

Adjacent sequences: A175194 A175195 A175196 * A175198 A175199 A175200

Keyword

easy,nonn,tabl

Author

R. J. Mathar, Mar 02 2010

Status

approved