A129173 - OEIS

%I #26 Jan 10 2022 10:24:17

%S 0,1,9,58,336,1853,9945,52487,273939,1418567,7303791,37441560,

%T 191287254,974642943,4955123955,25146686730,127424717400,644873878895,

%U 3260055588615,16465301636090,83092583965020,419031686115875

%N Total area below 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="/A129173/b129173.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)

%H E. Deutsch, E. Munarini and 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.

%H Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, <a href="https://doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562, Table 1.

%F a(n) = Sum_{k=0,..,n^2} k*A129172(n,k).

%F a(n) - 5*a(n-1) = A002212(n) + A002212(n-1).

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

%F (n+1)(n-2)a(n)-(11n^2-20n-6)a(n-1)+5(7n^2-19n+7)a(n-2)-25(n-1)(n-3)a(n-3) = 0.

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

%e a(2)=9 because the areas below the skew Dyck paths UDUD, UUDD and UUDL are 2, 4 and 3, respectively.

%p a[0]:=1: a[1]:=1: a[2]:=9: for n from 3 to 25 do a[n]:=((11*n^2-20*n-6)*a[n-1]-5*(7*n^2-19*n+7)*a[n-2]+25*(n-1)*(n-3)*a[n-3])/(n+1)/(n-2) od: seq(a[n],n=0..25);

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

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

%Y Cf. A002212, A129172.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 09 2007