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.

2, 4, 2, 6, 12, 2, 8, 30, 28, 2, 10, 56, 126, 60, 2, 12, 90, 344, 462, 124, 2, 14, 132, 730, 1880, 1566, 252, 2, 16, 182, 1332, 5370, 9368, 5070, 508, 2, 18, 240, 2198, 12372, 36250, 43736, 15966, 1020, 2, 20, 306, 3376, 24710, 106452, 228090, 195224, 49422, 2044, 2

Offset: 1

Author

Shel Kaphan, Mar 09 2024

Keywords

• tabl
• nonn

Comments

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.

Examples

			distance
 k      1   2    3      4       5        6         7          8
dims ----------------------------------------------------------
 n 1 |  2   2    2      2       2        2         2          2
   2 |  4  12   28     60     124      252       508       1020
   3 |  6  30  126    462    1566     5070     15966      49422
   4 |  8  56  344   1880    9368    43736    195224     844760
   5 | 10  90  730   5370   36250   228090   1359130    7771770
   6 | 12 132 1332  12372  106452   856212   6505812   47189652
   7 | 14 182 2198  24710  259574  2562182  23928758  213041990
   8 | 16 240 3376  44592  554416  6511920  72592816  772172592
		

Crossrefs

Columns: A002939 (k=2).

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

Formula

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