A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 4, 3, 3, 3, 1, 1, 1, 1, 3, 3, 5, 6, 8, 9, 11, 12, 11, 10, 7, 6, 4, 1, 1, 1, 1, 3, 3, 6, 6, 9, 12, 16, 18, 22, 27, 29, 33, 38, 40, 39, 39, 34, 28, 21, 14, 10, 5, 1, 1, 1, 1, 3, 3, 6, 6, 10, 13, 18, 22, 28, 35, 41, 50, 61

Offset: 0

Author

N. J. A. Sloane, Nov 26 2011

Keywords

• nonn
• tabf

Examples

			Triangle begins
1
1 1
1 1 1 2 1
1 1 1 3 3 4 3 3 3 1
1 1 1 3 3 5 6 8 9 11 12 11 10 7 6 4 1
...
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953. See Example 3.

Crossrefs

Row sums give A133656.

Programs

Maple

b:= proc(x, y) option remember; expand(`if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y)*z^(2*y)+b(x, y-1)+`if`(y>0, add(
       b(x-(2*j-1), y-1)*z^((2*y-1)*(2*j-1)), j=1..1+(x-y)/2), 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Feb 02 2018

Mathematica

b[x_, y_] := b[x, y] = Expand[If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y]*z^(2*y) + b[x, y - 1] + If[y > 0, Sum[b[x - (2*j - 1), y - 1]*z^((2*y - 1)*(2*j - 1)), {j, 1, 1 + (x - y)/2}], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, n]];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Extensions

More term from Alois P. Heinz, Feb 02 2018