A128748 Number of peaks at height >1 in all skew Dyck paths of semilength n.

0, 2, 11, 54, 260, 1247, 5982, 28741, 138364, 667488, 3226503, 15625476, 75802578, 368316888, 1792203759, 8732274312, 42598366616, 208036945958, 1017023261529, 4976560342522, 24372741339016, 119461561111023, 585970198529224

Offset: 1

Author

Emeric Deutsch, Mar 31 2007

Keywords

• nonn

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

Examples

			a(2)=2 because in the paths UDUD, U(UD)D and U(UD)L we have altogether 2 peaks at height >1 (shown between parentheses).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Crossrefs

Cf. A128747.

Programs

Maple

G:=(1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/2/z/(2-z)/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);

Mathematica

Rest[CoefficientList[Series[(1-4*x+2*x^2+x^3-(1-x+x^2)*Sqrt[1-6*x+5*x^2]) /2/x/(2-x)/Sqrt[1-6*x+5*x^2], {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

PARI

z='z+O('z^50); concat([0], Vec((1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017

Formula

a(n) = Sum_{k=0..n-1} A128747(n,k).

G.f.: (1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-z)*sqrt(1-6*z+5*z^2)).

a(n) ~ 5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence 2*(n+2)*a(n) +(-19*n-18)*a(n-1) +(53*n-12)*a(n-2) +2*(-20*n+19)*a(n-3) +(-n+26)*a(n-4) +5*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 17 2016