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A212382 Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
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, 5, 42, 1, 1, 1, 1, 1, 2, 12, 132, 1, 1, 1, 1, 1, 1, 6, 30, 429, 1, 1, 1, 1, 1, 1, 2, 16, 79, 1430, 1, 1, 1, 1, 1, 1, 1, 7, 37, 213, 4862, 1, 1, 1, 1, 1, 1, 1, 2, 22, 83, 584, 16796, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Lengths of descents are unrestricted.

For p>0 is column p asymptotic to a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where r and s are real roots (0 < r < 1) of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014

LINKS

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

Vaclav Kotesovec, Asymptotic of subsequences of A212382

FORMULA

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

G.f. of column k>0 is series_reversion(B(x))/x where B(x) = x/(1 + x + x^(k+1) + x^(2*k+1) + x^(3*k+1) + ... ) = x/(1+x/(1-x^k)); for Dyck paths with allowed ascent lengths {u_1, u_2, ...} use B(x) = x/( 1 + sum(k>=1, x^{u_k} ) ). - Joerg Arndt, Apr 23 2016

EXAMPLE

A(0,k) = 1: the empty path.

A(3,0) = 1: UDUDUD.

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

A(3,2) = 2: UDUDUD, UUUDDD.

A(5,3) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, 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,  5,  2,  1,  1,  1,  1, ...

  1,  42, 12,  6,  2,  1,  1,  1, ...

  1, 132, 30, 16,  7,  2,  1,  1, ...

  1, 429, 79, 37, 22,  8,  2,  1, ...

MAPLE

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

      `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, k, true)+

      `if`(u, add(b(x-(k*t+1), y, k, false), t=0..(x-1)/k), 0)))

    end:

A:= (n, k)-> `if`(k=0, 1, b(n, n, k, true)):

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

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

MATHEMATICA

b[x_, y_, k_, u_] := b[x, y, k, u] = If[x<0 || y<x, 0, If[x == 0 && y == 0, 1, b[x, y-1, k, True] + If[u, Sum[b[x-(k*t+1), y, k, False], {t, 0, (x-1)/k}], 0]]]; A[n_, k_] := If[k == 0, 1, b[n, n, k, True]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from first Maple program *)

CROSSREFS

Columns k=0-10 give: A000012, A000108, A101785, A212383, A212384, A212385, A212386, A212387, A212388, A212389, A212390.

A(2n,n) gives A323229.

Sequence in context: A119326 A219866 A212363 * A274835 A275069 A181937

Adjacent sequences:  A212379 A212380 A212381 * A212383 A212384 A212385

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 12 2012

STATUS

approved

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Last modified August 25 21:00 EDT 2019. Contains 326324 sequences. (Running on oeis4.)