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.
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
Offset: 0
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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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-François Alcover, Jan 15 2014, translated from first Maple program *)
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