OFFSET
5,1
COMMENTS
Triangle T(n,k), 3<=k<=n-2, given by (n-k)^k-2*C(n-1,k)+(n-k) is derived using inclusion/exclusion. The triangle contains several other listed sequences: T(2n,n) is sequence A056174(n), number of monotonic functions from [n] to [n]; T(n+2,n) is sequence A005803(n), second-order Eulerian numbers; and T(n,3) is A006331(n-4), maximum accumulated number of electrons at energy level n.
LINKS
Reinhard Zumkeller, Rows n=5..100 of triangle, flattened
Dennis Walsh, Notes on finite monotonic and non-monotonic functions
FORMULA
EXAMPLE
Triangle T(n,k) begins
n\k 3 4 5 6 7 8 9
5 2
6 10 8
7 28 54 22
8 60 190 204 52
9 110 490 916 676 114
10 182 1050 2878 3932 2118 240
11 280 1988 7278 15210 16148 6474 494
...
For n=6 and k=4, T(6,4)=8 since there are 8 non-monotonic functions f from [4] to [2], namely, f = <f(1),f(2),f(3),f(4)> given by <1,1,2,1>, <1,2,1,1>, <1,2,2,1>, <1,2,1,2>, <2,2,1,2>, <2,1,2,2>, <2,1,1,2>, and <2,1,2,1>.
MAPLE
seq(seq((n-k)^k-2*binomial(n-1, k)+(n-k), k=3..(n-2)), n=5..15);
MATHEMATICA
nmax = 15; t[n_, k_] := (n-k)^k-2*Binomial[n-1, k]+(n-k); Flatten[ Table[ t[n, k], {n, 5, nmax}, {k, 3, n-2}]](* Jean-François Alcover, Nov 18 2011, after Maple *)
PROG
(Haskell)
a189711 n k = (n - k) ^ k - 2 * a007318 (n - 1) k + n - k
a189711_row n = map (a189711 n) [3..n-2]
a189711_tabl = map a189711_row [5..]
-- Reinhard Zumkeller, May 16 2014
CROSSREFS
KEYWORD
AUTHOR
Dennis P. Walsh, Apr 25 2011
STATUS
approved