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A129179 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2). A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. 2
1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 4, 3, 2, 1, 1, 1, 1, 4, 6, 7, 10, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 1, 1, 1, 5, 10, 14, 21, 28, 31, 33, 34, 34, 31, 27, 25, 22, 17, 14, 13, 10, 7, 5, 4, 3, 2, 1, 1, 1, 1, 6, 15, 25, 40, 60, 77, 92, 106, 117, 122, 121, 120, 116, 107, 98, 91, 82, 71, 62, 54, 45 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has 1+n^2 terms. Row sums are the large Schroeder numbers (A006318). Sum(k*T(n,k), k>=0) = A129180(n).

LINKS

Alois P. Heinz, Rows n = 0..32, flattened

FORMULA

G.f.: G(t,z) satisfies G(t,z) = 1 + z*G(t,z) + t*z*G(t,t^2*z)*G(t,z).

EXAMPLE

T(3,5) = 3 because we have UDUUDD, UUDDUD and UHHD.

Triangle starts:

1;

1,1;

1,2,1,1,1;

1,3,3,3,4,3,2,1,1,1;

1,4,6,7,10,11,10,9,8,7,5,4,3,2,1,1,1;

MAPLE

G:=1/(1-z-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i)*z-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G, z=0, 13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 6 do seq(coeff(P[n], t, j), j=0..n^2) od; # yields sequence in triangular form

# second Maple program:

b:= proc(x, y) option remember; `if`(y>x or y<0, 0,

      `if`(x=0, 1, expand(b(x-1, y-1)*z^(y-1/2)

      +b(x-2, y)*z^(2*y) +b(x-1, y+1)*z^(y+1/2))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0)):

seq(T(n), n=0..7);  # Alois P. Heinz, May 27 2015

MATHEMATICA

b[x_, y_] := b[x, y] = If[y>x || y<0, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2)  + b[x-2, y]*z^(2*y) + b[x-1, y+1]*z^(y+1/2)]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0]]; Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A006318, A129180.

Sequence in context: A226444 A196929 A258445 * A120621 A201080 A039754

Adjacent sequences:  A129176 A129177 A129178 * A129180 A129181 A129182

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Apr 08 2007

STATUS

approved

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Last modified March 27 06:59 EDT 2017. Contains 284144 sequences.