A129156 - OEIS

%I #13 Feb 10 2017 01:11:35

%S 0,1,3,10,36,136,532,2139,8796,36859,156946,677514,2959669,13063493,

%T 58184838,261230814,1181144792,5374078726,24588562675,113067256235,

%U 522270436044,2422244159067,11275548912967,52663412854571

%N Number of primitive Dyck factors 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. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.

%H G. C. Greubel, <a href="/A129156/b129156.txt">Table of n, a(n) for n = 0..1000</a>

%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*A129154(n,k).

%F a(n) = A128742(n) - A129158(n).

%F G.f.: (3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2.

%F a(n) ~ (5-sqrt(5)) * 5^(n+3/2) / (36*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014

%e a(2)=3 because in all skew Dyck paths of semilength 3, namely (UD)(UD), (UUDD) and UUDL, we have altogether 3 primitive Dyck factors (shown between parentheses).

%p G:=(3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(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[(3-3*x-Sqrt[1-6*x+5*x^2])*(1-Sqrt[1-4*x])/ (1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)

%o (PARI) z='z+O('z^25); concat([0], Vec((3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2)) \\ _G. C. Greubel_, Feb 09 2017

%Y Cf. A129154, A129157, A129158.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Apr 02 2007