OFFSET
0,3
COMMENTS
Conjecture: Generally, column k is asymptotic to c(k) * d^n * n^(k-3/2), where d = 3.8821590268628506747194368909643384... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c(k) are specific constants (independent on n). - Vaclav Kotesovec, Jun 05 2014
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
EXAMPLE
T(4,1) = 1: UDUUDUDD.
T(5,1) = 5: UDUDUUDUDD, UDUUDUDDUD, UDUUDUDUDD, UDUUDUUDDD, UUDUUDUDDD.
T(6,1) = 19: UDUDUDUUDUDD, UDUDUUDUDDUD, UDUDUUDUDUDD, UDUDUUDUUDDD, UDUUDUDDUDUD, UDUUDUDDUUDD, UDUUDUDUDDUD, UDUUDUDUDUDD, UDUUDUDUUDDD, UDUUDUUDDDUD, UDUUDUUDDUDD, UDUUDUUUDDDD, UUDDUDUUDUDD, UUDUDUUDUDDD, UUDUUDUDDDUD, UUDUUDUDDUDD, UUDUUDUDUDDD, UUDUUDUUDDDD, UUUDUUDUDDDD.
T(6,2) = 1: UDUUDUUDUDDD.
T(7,2) = 7: UDUDUUDUUDUDDD, UDUUDUDUUDUDDD, UDUUDUUDUDDDUD, UDUUDUUDUDDUDD, UDUUDUUDUDUDDD, UDUUDUUDUUDDDD, UUDUUDUUDUDDDD.
T(8,3) = 1: UDUUDUUDUUDUDDDD.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 5;
: 4 : 13, 1;
: 5 : 37, 5;
: 6 : 112, 19, 1;
: 7 : 352, 70, 7;
: 8 : 1136, 259, 34, 1;
: 9 : 3742, 962, 149, 9;
: 10 : 12529, 3585, 627, 54, 1;
MAPLE
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 5, 2, 4][t])*
`if`(t=6, z, 1) +b(x-1, y-1, [1, 3, 1, 3, 6, 1][t]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..20);
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 2, 4}[[t]]]*If[t == 6, z, 1] + b[x-1, y-1, {1, 3, 1, 3, 6, 1}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 03 2014
STATUS
approved