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A201080 Irregular triangle read by rows: number of shifted Schroeder paths of length n and area k. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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.

EXAMPLE

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

...

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-Fran├žois Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

Row sums give A133656.

Sequence in context: A258445 A129179 A120621 * A039754 A213919 A062277

Adjacent sequences:  A201077 A201078 A201079 * A201081 A201082 A201083

KEYWORD

nonn,tabf,changed

AUTHOR

N. J. A. Sloane, Nov 26 2011

EXTENSIONS

More term from Alois P. Heinz, Feb 02 2018

STATUS

approved

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Last modified June 25 07:07 EDT 2018. Contains 311891 sequences. (Running on oeis4.)