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%I #49 Apr 28 2024 11:30:14
%S 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,
%T 480,384,128,2,28,168,560,1120,1344,896,256,2,32,224,896,2240,3584,
%U 3584,2048,512,2,36,288,1344,4032,8064,10752,9216,4608,1024
%N Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.
%C 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.
%C 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.)
%C T(2*n,3*n) = T(2*n+1,3*n), and this is A036909.
%F T(n,k) = binomial(k-1,n-1)*2^n.
%F G.f.: 2*x*y/(1 - y - 2*x*y). - _Stefano Spezia_, Apr 27 2024
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e -----------------------------------------------------------------------
%e 1 | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
%e 2 | 4 8 12 16 20 24 28 32 36 40 44 48 52 56
%e 3 | 8 24 48 80 120 168 224 288 360 440 528 624 728
%e 4 | 16 64 160 320 560 896 1344 1920 2640 3520 4576 5824
%e 5 | 32 160 480 1120 2240 4032 6720 10560 15840 22880 32032
%e 6 | 64 384 1344 3584 8064 16128 29568 50688 82368 128128
%e 7 | 128 896 3584 10752 26880 59136 118272 219648 384384
%e 8 | 256 2048 9216 30720 84480 202752 439296 878592
%e 9 | 512 4608 23040 84480 253440 658944 1537536
%e 10 | 1024 10240 56320 225280 732160 2050048
%e 11 | 2048 22528 135168 585728 2050048
%e 12 | 4096 49152 319488 1490944
%e 13 | 8192 106496 745472
%e 14 | 16384 229376
%e 15 | 32768
%e 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).
%t T[n_,k_]:=Binomial[k-1,n-1]*2^n; Table[T[n,k],{k,10},{n,k}]//Flatten
%o (Python)
%o from math import comb
%o def A370469_T(n,k): return comb(k-1,n-1)<<n # _Chai Wah Wu_, Apr 25 2024
%Y Cf. A033996, A333714 (n=3)
%Y Cf. A102860 (n=4).
%Y Cf. A036289, A097064, A134401 (+1-diagonal).
%Y Cf. A001815 (+2-diagonal).
%Y Cf. A371064.
%Y Cf. A036909.
%Y 2 * A013609.
%K tabl,nonn
%O 1,1
%A _Shel Kaphan_, Mar 30 2024