A129169 - OEIS

%I #17 Feb 10 2017 01:11:50

%S 0,0,2,9,37,153,643,2745,11881,52038,230268,1028007,4625013,20949444,

%T 95462022,437318631,2012933893,9304997544,43179620542,201076887771,

%U 939358727203,4401177906249,20676208303727,97374640613139

%N Number of UU's starting at level 0 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="/A129169/b129169.txt">Table of n, a(n) for n = 0..1000</a>(terms 0..200 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,..,floor(n/2)} k*A129168(n,k).

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

%F Recurrence: 2*n*(n+2)*a(n) = (13*n^2+8*n+4)*a(n-1) - (16*n^2-7*n-18)*a(n-2) + 5*(n-2)*(n+1)*a(n-3) . - _Vaclav Kotesovec_, Oct 20 2012

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

%e a(3)=9 because in the 10 skew Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, UD(UU)DL, (UU)DDUD, (UU)DUDD, (UU)UDDD, (UU)UDLD, (UU)DUDL, (UU)UDDL and (UU)UDLL, we have altogether 9 UU's starting at level 0 (shown between parentheses).

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

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

%o (PARI) z='z+O('z^25); concat([0,0], Vec(2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^2)) \\ _G. C. Greubel_, Feb 09 2017

%Y Cf. A129169.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 05 2007