OFFSET
0,7
COMMENTS
In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
Triangle T(n,k) begins:
00: 1;
01: 1, 0;
02: 1, 1, 0;
03: 2, 2, 1, 0;
04: 5, 6, 3, 1, 0;
05: 16, 20, 12, 4, 1, 0;
06: 61, 80, 50, 20, 5, 1, 0;
07: 271, 366, 240, 100, 30, 6, 1, 0;
08: 1372, 1897, 1281, 560, 175, 42, 7, 1, 0;
09: 7795, 10976, 7588, 3416, 1120, 280, 56, 8, 1, 0;
10: 49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0;
...
The 15 ascent sequences of length 4 (dots denote zeros) with their number of flat steps are:
01: [ . . . . ] 3
02: [ . . . 1 ] 2
03: [ . . 1 . ] 1
04: [ . . 1 1 ] 2
05: [ . . 1 2 ] 1
06: [ . 1 . . ] 1
07: [ . 1 . 1 ] 0
08: [ . 1 . 2 ] 0
09: [ . 1 1 . ] 1
10: [ . 1 1 1 ] 2
11: [ . 1 1 2 ] 1
12: [ . 1 2 . ] 0
13: [ . 1 2 1 ] 0
14: [ . 1 2 2 ] 1
15: [ . 1 2 3 ] 0
There are 5 sequences without flat steps, 6 with one flat step, etc., giving row [5, 6, 3, 1, 0] for n=4.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
`if`(j=i, x, 1) *b(n-1, j, t+`if`(j>i, 1, 0)), j=0..t+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, -1$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == i, x, 1]*b[n-1, j, t + If[j>i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][ b[n, -1, -1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, May 05 2014
STATUS
approved