A370469 - OEIS

A370469

Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.

2, 2, 4, 2, 8, 8, 2, 12, 24, 16, 2, 16, 48, 64, 32, 2, 20, 80, 160, 160, 64, 2, 24, 120, 320, 480, 384, 128, 2, 28, 168, 560, 1120, 1344, 896, 256, 2, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 2, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

T(n,k) is the number of points on the n-dimensional cross polytope with facets at distance k from the origin which have no coordinate equal to 0.

T(n,n) = 2^n. The (n-1)-dimensional simplex at distance n from the origin in Z^n has exactly 1 point with no zero coordinates, at (1,1,...,1). There are 2^n (n-1)-dimensional simplexes at distance n from the origin as part of the cross polytope in Z^n. (The lower dimensional facets do not count as they have at least one 0 coordinate.)

T(2*n,3*n) = T(2*n+1,3*n), and this is A036909.

LINKS

Table of n, a(n) for n=1..55.

FORMULA

T(n,k) = binomial(k-1,n-1)*2^n.

G.f.: 2*x*y/(1 - y - 2*x*y). - Stefano Spezia, Apr 27 2024

EXAMPLE

n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-----------------------------------------------------------------------

1 | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 | 4 8 12 16 20 24 28 32 36 40 44 48 52 56

3 | 8 24 48 80 120 168 224 288 360 440 528 624 728

4 | 16 64 160 320 560 896 1344 1920 2640 3520 4576 5824

5 | 32 160 480 1120 2240 4032 6720 10560 15840 22880 32032

6 | 64 384 1344 3584 8064 16128 29568 50688 82368 128128

7 | 128 896 3584 10752 26880 59136 118272 219648 384384

8 | 256 2048 9216 30720 84480 202752 439296 878592

9 | 512 4608 23040 84480 253440 658944 1537536

10 | 1024 10240 56320 225280 732160 2050048

11 | 2048 22528 135168 585728 2050048

12 | 4096 49152 319488 1490944

13 | 8192 106496 745472

14 | 16384 229376

15 | 32768

The cross polytope in Z^3 (the octahedron) with points at distance 3 from the origin has 8 triangle facets, each with edge length 4. There is one point in the center of each triangle with coordinates (+-1,+-1,+-1).

MATHEMATICA

T[n_, k_]:=Binomial[k-1, n-1]*2^n; Table[T[n, k], {k, 10}, {n, k}]//Flatten

PROG

(Python)

from math import comb

def A370469_T(n, k): return comb(k-1, n-1)<<n # Chai Wah Wu, Apr 25 2024

CROSSREFS

Cf. A033996, A333714 (n=3)

Cf. A102860 (n=4).

Cf. A036289, A097064, A134401 (+1-diagonal).

Cf. A001815 (+2-diagonal).

Cf. A371064.