A129171 - OEIS

%I #18 Feb 10 2017 01:12:00

%S 0,1,6,32,165,840,4251,21443,107946,542680,2725635,13679997,68623176,

%T 344090307,1724754180,8642952000,43300971885,216895107480,

%U 1086253033035,5439405705125,27234492215400,136345625309965,682531666024170

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

%C 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.

%H G. C. Greubel and Vincenzo Librandi, <a href="/A129171/b129171.txt">Table of n, a(n) for n = 0..1000</a>(terms 1..300 from Vincenzo Librandi)

%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

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

%F 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

%F 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

%F a(n) ~ 3*5^(n-1)/2*(1-sqrt(5)/(6*sqrt(Pi*n))) . - _Vaclav Kotesovec_, Oct 20 2012

%e 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.

%p 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);

%t 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 *)

%o (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

%Y Cf. A129170.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 07 2007