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A212363 Number A(n,k) of Dyck n-paths all of whose ascents and descents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 14, 1, 1, 1, 1, 1, 4, 42, 1, 1, 1, 1, 1, 2, 8, 132, 1, 1, 1, 1, 1, 1, 4, 17, 429, 1, 1, 1, 1, 1, 1, 2, 7, 37, 1430, 1, 1, 1, 1, 1, 1, 1, 4, 12, 82, 4862, 1, 1, 1, 1, 1, 1, 1, 2, 7, 22, 185, 16796, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FORMULA

G.f. of column k>0 satisfies: A_k(x) = 1+A_k(x)*(x-x^k*(1-A_k(x))), g.f. of column k=0: A_0(x) = 1/(1-x).

A(n,k) = A(n-1,k) + Sum_{j=1..n-k} A(j,k)*A(n-k-j,k) for n,k>0; A(n,0) = A(0,k) = 1.

G.f. of column k > 0: (1 - x + x^k - sqrt((1 - x + x^k)^2 - 4*x^k)) / (2*x^k). - Vaclav Kotesovec, Sep 02 2014

EXAMPLE

A(3,0) = 1: UDUDUD.

A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.

A(4,2) = 4: UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD.

A(5,2) = 8: UDUDUDUDUD, UDUDUUUDDD, UDUUUDDDUD, UDUUUDUDDD, UUUDDDUDUD, UUUDUDDDUD, UUUDUDUDDD, UUUUUDDDDD.

A(5,3) = 4: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDUDDDD.

Square array A(n,k) begins:

  1,   1,  1,  1,  1,  1,  1,  1, ...

  1,   1,  1,  1,  1,  1,  1,  1, ...

  1,   2,  1,  1,  1,  1,  1,  1, ...

  1,   5,  2,  1,  1,  1,  1,  1, ...

  1,  14,  4,  2,  1,  1,  1,  1, ...

  1,  42,  8,  4,  2,  1,  1,  1, ...

  1, 132, 17,  7,  4,  2,  1,  1, ...

  1, 429, 37, 12,  7,  4,  2,  1, ...

MAPLE

A:= proc(n, k) option remember;

      `if`(k=0, 1, `if`(n=0, 1, A(n-1, k)

                   +add(A(j, k)*A(n-k-j, k), j=1..n-k)))

    end:

seq(seq(A(n, d-n), n=0..d), d=0..15);

# second Maple program:

A:= (n, k)-> `if`(k=0, 1, coeff(series(RootOf(

              A||k=1+A||k*(x-x^k*(1-A||k)), A||k), x, n+1), x, n)):

seq(seq(A(n, d-n), n=0..d), d=0..15);

MATHEMATICA

A[n_, k_] := A[n, k] = If[k == 0, 1, If[n == 0, 1, A[n-1, k] + Sum[A[j, k]*A[n-k-j, k], {j, 1, n-k}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, Jan 15 2014, translated from first Maple program *)

CROSSREFS

Columns k=0-10 give: A000012, A000108, A004148, A023432, A023427, A212364, A212365, A212366, A212367, A212368, A212369.

Sequence in context: A174547 A119326 A219866 * A212382 A274835 A275069

Adjacent sequences:  A212360 A212361 A212362 * A212364 A212365 A212366

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified March 26 14:18 EDT 2019. Contains 321497 sequences. (Running on oeis4.)