OFFSET
0,5
LINKS
Mike Earnest, Counting endofunctions by inclusion-exclusion, at Math.StackExchange.
FORMULA
Sum_{k=0..n} (-1)^(n - k)*T(n, k) = n^n. - Werner Schulte, Jun 03 2022 in A000312. [Formerly a conjecture, now proved by Mike Earnest, see link.]
EXAMPLE
Table T(n, k) begins:
[0] 1
[1] 0, 1
[2] 0, 3, 7
[3] 0, 12, 75, 90
[4] 0, 60, 715, 2100, 1701
[5] 0, 360, 7000, 36750, 69510, 42525
[6] 0, 2520, 72884, 595350, 1940295, 2692305, 1323652
[7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280
MAPLE
T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jun 06 2022
STATUS
approved