cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

Original entry on oeis.org

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

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Author

Alois P. Heinz, May 10 2012

Keywords

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, ...
		

Crossrefs

Programs

  • 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