OFFSET
0,2
COMMENTS
a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).
LINKS
H. Mühle, Counting Proper Mergings of Chains and Antichains, Discrete Math., Vol. 327(C), 2014, 118-129. Also arXiv:1206.3922 [math.CO], 2012.
FORMULA
a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.
EXAMPLE
For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.
MAPLE
a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):
seq(a(n), n=0..16); # Peter Luschny, Nov 11 2016
MATHEMATICA
a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)
PROG
(PARI) a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ Michel Marcus, Jul 14 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Henri Mühle, Nov 11 2016
STATUS
approved