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%I #21 Feb 21 2024 01:11:59
%S 0,0,0,0,1,2,7,14,37,74,176,352,794,1588,3473,6946,14893,29786,63004,
%T 126008,263950,527900,1097790,2195580,4540386,9080772,18696432,
%U 37392864,76717268,153434536,313889477,627778954,1281220733,2562441466,5219170052,10438340104
%N Number of valleys at level 0 in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0) steps at positive heights).
%H G. C. Greubel, <a href="/A191389/b191389.txt">Table of n, a(n) for n = 0..1000</a>
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2402.13026">Dispersed Dyck paths revisited</a>, arXiv:2402.13026 [math.CO], 2024.
%F a(n) = Sum_{k=0..n} k*A191387(n,k).
%F G.f.: 2*(1-2*z^2-sqrt(1-4*z^2))/(1-2*z+sqrt(1-4*z^2))^2.
%F a(n) ~ 2^(n-1) * (1-3*sqrt(2)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F D-finite with recurrence -(n+2)*(n-4)*a(n) +2*(n+2)*(n-4)*a(n-1) +4*(n-2)*(n-3)*a(n-2) -8*(n-2)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Sep 24 2021
%e a(5)=2 because in HHHHH, HHHUD, HHUDH, HUDHH, HUUDD, UDHHH, UDHUD, UUDDH, HUDUD, and UDUDH only the last 2 paths have 1 valley at level 0; here U=(1,1), D=(1,-1), H=(1,0).
%p g := (2*(1-2*z^2-sqrt(1-4*z^2)))/(1-2*z+sqrt(1-4*z^2))^2: gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%t CoefficientList[Series[(2*(1-2*x^2-Sqrt[1-4*x^2]))/(1-2*x+Sqrt[1-4*x^2])^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) z='z+O('z^50); concat([0,0,0,0], Vec(2*(1-2*z^2 -sqrt(1-4*z^2)) /(1 - 2*z + sqrt(1-4*z^2))^2)) \\ _G. C. Greubel_, Feb 12 2017
%Y Cf. A191387. Convolution square of A037952 (shifted 2 places right).
%K nonn
%O 0,6
%A _Emeric Deutsch_, Jun 02 2011
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