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

0,4

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

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A024482, A097613.

Sequence in context: A184309 A051944 A153373 * A371871 A166677 A318062

Adjacent sequences: A225012 A225013 A225014 * A225016 A225017 A225018

Keyword

nonn

Author

David Scambler, Apr 23 2013

Status

approved