OFFSET
1,1
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.
FORMULA
T(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * 2^i * i^k.
EXAMPLE
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
KEYWORD
tabl,nonn
AUTHOR
Shel Kaphan, Mar 09 2024
STATUS
approved