OFFSET
0,3
COMMENTS
Terms in column k are multiples of k!.
LINKS
Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
Wikipedia, Permutation
FORMULA
T(3n,n) = n!.
EXAMPLE
T(4,1) = 4: 1342, 2314, 3421, 4231.
Triangle T(n,k) begins:
1;
1;
2;
5, 1;
20, 4;
102, 18;
626, 92, 2;
4458, 564, 18;
36144, 4032, 144;
328794, 32898, 1182, 6;
3316944, 301248, 10512, 96;
36755520, 3057840, 102240, 1200;
443828184, 34073184, 1085904, 14304, 24;
5800823880, 413484240, 12538080, 174000, 600;
81591320880, 5428157760, 156587040, 2214720, 10800;
1228888215960, 76651163160, 2105035440, 29777520, 175800, 120;
...
MAPLE
b:= proc(s, l, t) option remember; `if`(s={}, 1, add((h->
expand(b(s minus {j}, j, `if`(h=1, 2, 1))*
`if`(t=2 and h=-2, x, 1)))(j-l), j=s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
b({$1..n}, -1, 1)):
seq(T(n), n=0..13);
MATHEMATICA
b[s_, l_, t_] := b[s, l, t] = If[s == {}, 1, Sum[Function[h,
Expand[b[s ~Complement~ {j}, j, If[h == 1, 2, 1]]*
If[t == 2 && h == -2, x, 1]]][j - l], {j, s}]];
T[n_] := CoefficientList[b[Range[n], -1, 1], x];
T /@ Range[0, 13] // Flatten (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Apr 18 2021
STATUS
approved