A225015 Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.

0, 1, 1, 5, 18, 66, 245, 918, 3465, 13156, 50193, 192270, 739024, 2848860, 11009778, 42642460, 165480975, 643281480, 2504501625, 9764299710, 38115568260, 148955040300, 582714871830, 2281745337300, 8942420595810, 35074414899576, 137672461877850, 540756483094828

Offset: 0

Author

David Scambler, Apr 23 2013

Keywords

• nonn

Comments

A sawtooth pattern of length 1 is UD not followed by UD.

First differences of A024482.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A024482, A097613.

Programs

Magma

A024482:= func< n | (3*n-2)*Catalan(n-1)/2 >;
A225015:= func< n | n le 2 select Floor((n+1)/2) else A024482(n) - A024482(n-1) >;
[A225015(n): n in [0..40]]; // G. C. Greubel, Apr 03 2024

Maple

a:= proc(n) option remember; `if`(n<4, [0, 1, 1, 5][n+1],
       (n-1)*(3*n-4)*(4*n-10)*a(n-1)/(n*(n-2)*(3*n-7)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 24 2013

Mathematica

Join[{0, 0, 1}, Table[(Binomial[2n, n]-Binomial[2n-2, n-1])/2, {n, 2, 25}]] // Differences (* Jean-François Alcover, Nov 12 2020 *)

SageMath

def A024482(n): return (3*n-2)*catalan_number(n-1)/2
def A225015(n): return floor((n+1)/2) if n<3 else A024482(n) - A024482(n-1)
[A225015(n) for n in range(41)] # G. C. Greubel, Apr 03 2024

Formula

a(0)=0, a(1)=1, a(n) = A024482(n) - A024482(n-1) for n >= 2.

From G. C. Greubel, Apr 03 2024: (Start)

G.f.: (1-x)^2*(1 - sqrt(1-4*x))/(2*sqrt(1-4*x)).

E.g.f.: -(1/4)*(2-4*x+x^2) + (1/12)*Exp(2*x)*((6-12*x+43*x^2-24*x^3) *BesselI(0, 2*x) - 4*x*(7-5*x)*BesselI(1,2*x) - 3*x^2*(13-8*x)* BesselI(2,2*x)). (End)