OFFSET
0,4
COMMENTS
Conjecture: column k > 0 is asymptotic to n * 2^(n-2*k) * k^(n-1). - Vaclav Kotesovec, Oct 26 2015
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FindStat - Combinatorial Statistic Finder, The number of cyclic valleys and cyclic peaks of a permutation.
FORMULA
T(n,k) = n*A008303(n-1, k-1) for n > 1. - Andrew Howroyd, May 13 2020
EXAMPLE
Triangle begins:
1;
1;
0, 2;
0, 6;
0, 16, 8;
0, 40, 80;
0, 96, 528, 96;
...
MAPLE
b:= proc(u, o, t) option remember; expand(`if`(u+o=0, x,
add(b(u-j, o+j-1, 0), j=1..u)*`if`(min(t, n)>0, x, 1)+
add(b(u+j-1, o-j, 1), j=1..o)))
end:
T:= n-> `if`(n<2, 1, (p-> seq(n*coeff(p, x, i)
, i=0..degree(p)))(b(n-1, 0$2))):
seq(T(n), n=0..14); # Alois P. Heinz, Oct 28 2015
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = Expand[If[u+o == 0, x, Sum[b[u-j, o+j-1, 0], {j, 1, u}]*If[Min[t, n] > 0, x, 1] + Sum[b[u+j-1, o-j, 1], {j, 1, o}]]]; T[n_] := If[n<2, 1, Function[p, Table[n*Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Christian Stump, Oct 26 2015
EXTENSIONS
More terms from Alois P. Heinz, Oct 26 2015
STATUS
approved