OFFSET
1,3
COMMENTS
It appears that each column, other than the first, has asymptotic growth rate of 4.
FORMULA
G.f.: Sum_{n>=1, 1<=k<=n} T(n,k) x^n y^k = C(x) - 1 + ((1 - y) (1 - x y) (1 - (1 - x y)C(x)))/((1 - 2 x y) (1 - y + x y^2) ) where C(x) = 1 + x + 2x^2 + 5x^3 + ... is the g.f. for the Catalan numbers A000108 (conjectured).
EXAMPLE
For example, for the 123-avoiding permutation p = 42513, the 3 largest entries, 453, avoid 132 but the 4 largest entries, 4253, do not, and so p is counted by T(5,3).
Triangle begins:
1
0 2
0 1 4
0 3 3 8
0 9 10 7 16
0, 28, 32, 25, 15, 32
...
MATHEMATICA
Map[Rest, Rest[Map[CoefficientList[#, y] &, CoefficientList[ Normal[Series[ c - 1 + ((1 - y) (1 - x y) (1 - (1 - x y) c ))/((1 - 2 x y) (1 - y + x y^2)) /. {c :> (1 - Sqrt[1 - 4 x])/(2 x)}, {x, 0, 10}, {y, 0, 10}]], x]]]]
u[1, 1] = 1; u[2, 2] = 2;
u[n_, 1] /; n > 1 := 0; u[n_, k_] /; n < 1 || k < 1 || k > n := 0;
u[n_, k_] /; n >= 3 && 2 <= k <= n := u[n, k] = 3 u[n - 1, k - 1] - 2 u[n - 2, k - 2] + u[n, k + 1] - 2 u[n - 1, k] + If[k == 2, CatalanNumber[n - 2], 0];
Table[u[n, k], {n, 10}, {k, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David Callan, May 31 2016
STATUS
approved