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%I #12 Feb 10 2017 01:11:43
%S 0,0,1,5,22,96,422,1871,8360,37610,170222,774561,3541487,16263250,
%T 74981226,346957923,1610847944,7501970397,35038158569,164083453482,
%U 770312822822,3624741537711,17093452878067,80773023036909
%N Number of primitive non-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 non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.
%H G. C. Greubel, <a href="/A129158/b129158.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*A129157(n,k).
%F a(n) = A128742(n) - A129156(n).
%F G.f.: (1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2.
%F a(n) ~ (3*sqrt(5)+5) * 5^(1+n) / (36*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%e a(2)=1 because in all skew Dyck paths of semilength 3, namely UDUD, UUDD and (UUDL), we have altogether 1 primitive non-Dyck factor (shown between parentheses).
%p G:=(1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2)-sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27);
%t CoefficientList[Series[(1-5*x+3*(1-x)*Sqrt[1-4*x]-3*Sqrt[1-6*x+5*x^2]-Sqrt[(1-4*x)*(1-6*x+5*x^2)])/(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([0,0], Vec((1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2))) /(1+z+ sqrt(1-6*z+5*z^2) )^2)) \\ _G. C. Greubel_, Feb 09 2017
%Y Cf. A129157, A129156, A128742.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Apr 02 2007
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