A129171 Sum of the heights of the peaks in all skew Dyck paths of semilength n.

0, 1, 6, 32, 165, 840, 4251, 21443, 107946, 542680, 2725635, 13679997, 68623176, 344090307, 1724754180, 8642952000, 43300971885, 216895107480, 1086253033035, 5439405705125, 27234492215400, 136345625309965, 682531666024170

Offset: 0

Author

Emeric Deutsch, Apr 07 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)=6 because in the 3 skew Dyck paths of semilength 2, namely UDUD, UUDD and UUDL, the heights of the peaks are 1,1,2 and 2.
		

Links

• G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 1..300 from Vincenzo Librandi)
• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Cf. A129170.

Programs

Maple

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

Mathematica

CoefficientList[Series[x*(3 - 3*x - Sqrt[1 - 6*x + 5*x^2])/(1 - 6*x + 5*x^2)/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

z='z+O('z^25); concat([0], Vec(z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2)) \\ G. C. Greubel, Feb 10 2017

Formula

a(n) = Sum_{k=0,..,n} k*A129170(n,k).

G.f.: z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2. - corrected by Vaclav Kotesovec, Oct 20 2012

Recurrence: (n-1)*a(n) = (11*n-19)*a(n-1) - 5*(7*n-17)*a(n-2) + 25*(n-3)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3*5^(n-1)/2*(1-sqrt(5)/(6*sqrt(Pi*n))) . - Vaclav Kotesovec, Oct 20 2012