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.

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A097608 Triangle read by rows: number of Dyck paths of semilength n and having abscissa of the leftmost valley equal to k (if no valley, then it is taken to be 2n; 2<=k<=2n).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 3, 1, 1, 0, 1, 14, 9, 9, 4, 3, 1, 1, 0, 1, 42, 28, 28, 14, 10, 4, 3, 1, 1, 0, 1, 132, 90, 90, 48, 34, 15, 10, 4, 3, 1, 1, 0, 1, 429, 297, 297, 165, 117, 55, 35, 15, 10, 4, 3, 1, 1, 0, 1, 1430, 1001, 1001, 572, 407, 200, 125, 56, 35, 15, 10, 4, 3, 1, 1, 0, 1
Offset: 1

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Author

Emeric Deutsch, Aug 30 2004, Dec 22 2004

Keywords

Comments

A valley point is a path vertex that is preceded by a downstep and followed by an upstep (or by nothing at all). T(n,k) is the number of Dyck n-paths whose first valley point is at position k, 2<=k<=2n. - David Callan, Mar 02 2005
Row n has 2n-1 terms.
Row sums give the Catalan numbers (A000108).
Columns k=2 through 7 are respectively A000108, A000245, A071724, A002057, A071725, A026013. The nonzero entries in the even-indexed columns approach A088218 and similarly the odd-indexed columns approach A001791.

Examples

			Triangle begins
\ k..2...3...4...5...6...7....
n
1 |..1
2 |..1...0...1
3 |..2...1...1...0...1
4 |..5...3...3...1...1...0...1
5 |.14...9...9...4...3...1...1...0...1
6 |.42..28..28..14..10...4...3...1...1...0...1
7 |132..90..90..48..34..15..10...4...3...1...1...0...1
T(4,3)=3 because we have UU(DU)DDUD, UU(DU)DUDD and UU(DU)UDDD, where U=(1,1), D=(1,-1) (the first valley, with abscissa 3, is shown between parentheses).
		

Crossrefs

Programs

  • Maple
    G:=t^2*z*C*(1-t*z)/(1-t^2*z)/(1-t*z*C): C:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G,z=0,11)): for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: seq(seq(coeff(P[n],t^k),k=2..2*n),n=1..10);

Formula

G.f.=t^2*zC(1-tz)/[(1-t^2*z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
G.f. Sum_{2<=k<=2n}T(n, k)x^n*y^k = ((1 - (1 - 4*x)^(1/2))*y^2*(1 - x*y))/(2*(1 - ((1 - (1 - 4*x)^(1/2))*y)/2)*(1 - x*y^2)). With G:= (1 - (1 - 4*x)^(1/2))/2, the gf for column 2k is G(G^(2k+1)(G-x)-x^(k+1)(1-G))/(G^2-x) and for column 2k+1 is G(G-x)(G^(2k+2)-x^(k+1))/(G^2-x). - David Callan, Mar 02 2005

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 23 2007
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