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

0,3

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.

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A129170.

Sequence in context: A097139 A034942 A046714 * A082585 A084326 A199699

Adjacent sequences: A129168 A129169 A129170 * A129172 A129173 A129174

Keyword

nonn

Author

Emeric Deutsch, Apr 07 2007

Status

approved