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A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n). 10
1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 1, 0, 6, 0, 6, 0, 1, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1, 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1, 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1, 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1, 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,19

COMMENTS

Triangle A129182 transposed.

Column sums give the Catalan numbers (A000108).

Row sums give A143951.

Sums along falling diagonals give A005169.

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

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

G.f.: F(x,y) satisfies F(x,y) = 1 / (1 -  x*y * F(x, x^2*y) ).

G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

EXAMPLE

Triangle 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;

...

Column k=4 corresponds to the following 14 paths (dots denote zeros):

#:         path              area   steps (Dyck word)

01:  [ . 1 . 1 . 1 . 1 . ]     4     + - + - + - + -

02:  [ . 1 . 1 . 1 2 1 . ]     6     + - + - + + - -

03:  [ . 1 . 1 2 1 . 1 . ]     6     + - + + - - + -

04:  [ . 1 . 1 2 1 2 1 . ]     8     + - + + - + - -

05:  [ . 1 . 1 2 3 2 1 . ]    10     + - + + + - - -

06:  [ . 1 2 1 . 1 . 1 . ]     6     + + - - + - + -

07:  [ . 1 2 1 . 1 2 1 . ]     8     + + - - + + - -

08:  [ . 1 2 1 2 1 . 1 . ]     8     + + - + - - + -

09:  [ . 1 2 1 2 1 2 1 . ]    10     + + - + - + - -

10:  [ . 1 2 1 2 3 2 1 . ]    12     + + - + + - - -

11:  [ . 1 2 3 2 1 . 1 . ]    10     + + + - - - + -

12:  [ . 1 2 3 2 1 2 1 . ]    12     + + + - - + - -

13:  [ . 1 2 3 2 3 2 1 . ]    14     + + + - + - - -

14:  [ . 1 2 3 4 3 2 1 . ]    16     + + + + - - - -

There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is

[0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].

Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:

Semilength 4:

  [ . 1 . 1 2 1 2 1 . ]

  [ . 1 2 1 . 1 2 1 . ]

  [ . 1 2 1 2 1 . 1 . ]

Semilength 6:

  [ . 1 . 1 . 1 . 1 . 1 2 1 . ]

  [ . 1 . 1 . 1 . 1 2 1 . 1 . ]

  [ . 1 . 1 . 1 2 1 . 1 . 1 . ]

  [ . 1 . 1 2 1 . 1 . 1 . 1 . ]

  [ . 1 2 1 . 1 . 1 . 1 . 1 . ]

Semilength 8:

  [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

MAPLE

b:= proc(x, y, k) option remember;

      `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),

       b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))

    end:

T:= (n, k)-> b(2*k, 0, n):

seq(seq(T(n, k), k=0..n), n=0..20);  # Alois P. Heinz, Mar 29 2014

MATHEMATICA

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-Fran├žois Alcover, Feb 18 2015, after Alois P. Heinz *)

PROG

(PARI)

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }

print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }

N=20; x='x+O('x^N);

F(x, y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );

v= Vec( F(x, y) );

print_triangle(v)

CROSSREFS

Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))),  A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).

Cf. A047998, A138158, A227543.

Cf. A129181.

Sequence in context: A231642 A288318 A219483 * A069846 A239657 A298934

Adjacent sequences:  A239924 A239925 A239926 * A239928 A239929 A239930

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Mar 29 2014

STATUS

approved

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)