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A129182 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2). 9
1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 7, 0, 7, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 14, 0, 17, 0, 16, 0, 16, 0, 14, 0, 11, 0, 9, 0, 7, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

Row n has n^2 + 1 terms.

Row sums are the Catalan numbers (A000108).

Sum(k*T(n,k), k=0..n^2) = A008549(n).

Sums along falling diagonals give A005169. - Joerg Arndt, Mar 29 2014

T(2n,4n) = A240008(n). - Alois P. Heinz, Mar 30 2014

LINKS

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

FORMULA

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

Sum_{k=0..n^2} (n^2-k)/2 * T(n,k) = A139262(n). - Alois P. Heinz, Mar 31 2018

EXAMPLE

T(4,10) = 3 because we have UDUUUDDD, UUUDDDUD and UUDUDUDD.

Triangle starts:

1;

0,1;

0,0,1,0,1;

0,0,0,1,0,2,0,1,0,1;

0,0,0,0,1,0,3,0,3,0,3,0,2,0,1,0,1;

0,0,0,0,0,1,0,4,0,6,0,7,0,7,0,5,0,5,0,3,0,2,0,1,0,1;

Transposed triangle (A239927) begins:

00:  1;

01:  0, 1;

02:  0, 0, 1;

03:  0, 0, 0, 1;

04:  0, 0, 1, 0, 1;

05:  0, 0, 0, 2, 0, 1;

06:  0, 0, 0, 0, 3, 0, 1;

07:  0, 0, 0, 1, 0, 4, 0, 1;

08:  0, 0, 0, 0, 3, 0, 5, 0, 1;

09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;

10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;

11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;

12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;

13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;

14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;

15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;

16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;

17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;

18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;

19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;

20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;

... - Joerg Arndt, Mar 25 2014

MAPLE

G:=1/(1-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G, z=0, 11)): for n from 0 to 7 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 7 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<0 or y>x, 0, `if`(x=0, 1,

       expand(b(x-1, y-1)*z^(y-1/2)+ 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..10);  # Alois P. Heinz, Mar 29 2014

MATHEMATICA

b[x_, y_] := b[x, y] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y-1]*z^(y-1/2) + 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, 10}] // Flatten (* Jean-Fran├žois Alcover, Mar 24 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000108, A008549, A139262, A240008, A143951 (column sums).

Sequence in context: A323591 A105348 A016406 * A116857 A322338 A158971

Adjacent sequences:  A129179 A129180 A129181 * A129183 A129184 A129185

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Apr 08 2007

STATUS

approved

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Last modified February 20 20:55 EST 2019. Contains 320345 sequences. (Running on oeis4.)