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Array read by ascending antidiagonals where T(n,k) is the number of paths of length k from the origin to a facet of the cross polytope of size k in Z^n.
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%I #33 Mar 23 2024 21:12:47

%S 2,4,2,6,12,2,8,30,28,2,10,56,126,60,2,12,90,344,462,124,2,14,132,730,

%T 1880,1566,252,2,16,182,1332,5370,9368,5070,508,2,18,240,2198,12372,

%U 36250,43736,15966,1020,2,20,306,3376,24710,106452,228090,195224,49422,2044,2

%N Array read by ascending antidiagonals where T(n,k) is the number of paths of length k from the origin to a facet of the cross polytope of size k in Z^n.

%C In the cross polytope of dimension n, each facet of dimension i-1 (i=1..n) has i^k paths of length k from the origin to its surface, and there are binomial(n,i)*2^i such facets. To avoid double counting, an alternating sum is used to add up the paths to all the facets.

%F T(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * 2^i * i^k.

%e distance

%e k 1 2 3 4 5 6 7 8

%e dims ----------------------------------------------------------

%e n 1 | 2 2 2 2 2 2 2 2

%e 2 | 4 12 28 60 124 252 508 1020

%e 3 | 6 30 126 462 1566 5070 15966 49422

%e 4 | 8 56 344 1880 9368 43736 195224 844760

%e 5 | 10 90 730 5370 36250 228090 1359130 7771770

%e 6 | 12 132 1332 12372 106452 856212 6505812 47189652

%e 7 | 14 182 2198 24710 259574 2562182 23928758 213041990

%e 8 | 16 240 3376 44592 554416 6511920 72592816 772172592

%Y Columns: A002939 (k=2).

%Y Rows: A028399 (n=2), A366058 (n=3).

%K tabl,nonn

%O 1,1

%A _Shel Kaphan_, Mar 09 2024