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A371064 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.

%I A371064 #33 Mar 23 2024 21:12:47
%S A371064 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 A371064 1880,1566,252,2,16,182,1332,5370,9368,5070,508,2,18,240,2198,12372,
%U A371064 36250,43736,15966,1020,2,20,306,3376,24710,106452,228090,195224,49422,2044,2
%N A371064 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 A371064 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 A371064 T(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * 2^i * i^k.
%e A371064 distance
%e A371064  k      1   2    3      4       5        6         7          8
%e A371064 dims ----------------------------------------------------------
%e A371064  n 1 |  2   2    2      2       2        2         2          2
%e A371064    2 |  4  12   28     60     124      252       508       1020
%e A371064    3 |  6  30  126    462    1566     5070     15966      49422
%e A371064    4 |  8  56  344   1880    9368    43736    195224     844760
%e A371064    5 | 10  90  730   5370   36250   228090   1359130    7771770
%e A371064    6 | 12 132 1332  12372  106452   856212   6505812   47189652
%e A371064    7 | 14 182 2198  24710  259574  2562182  23928758  213041990
%e A371064    8 | 16 240 3376  44592  554416  6511920  72592816  772172592
%Y A371064 Columns: A002939 (k=2).
%Y A371064 Rows: A028399 (n=2), A366058 (n=3).
%K A371064 tabl,nonn
%O A371064 1,1
%A A371064 _Shel Kaphan_, Mar 09 2024