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%I #12 Jan 30 2020 21:29:18
%S 1,1,0,1,2,1,0,2,4,2,0,5,10,5,0,14,28,14,0,42,84,42,0,132,264,132,0,
%T 429,858,429,0,1430,2860,1430,0,4862,9724,4862,0,16796,33592,16796,0,
%U 58786,117572,58786,0,208012,416024,208012,0,742900,1485800,742900,0,2674440,5348880,2674440,0
%N Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except UH, HU, HD and DH.
%C The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
%F G.f.: ((1+t)(1+t-2t^4-(1+t)*sqrt(1-4t^4)))/(2t^5).
%F a(4n)=2*C(n), a(4n-1)=C(n), a(4n+1)=C(n) and a(4n+2)=0, where C(n) are the Catalan numbers A000108.
%F D-finite with recurrence: (n+5)*(3*n^2-27*n+92)*a(n) +16*(3*n-19)*a(n-1) +16*(-3*n+22)*a(n-2) +16*(3*n-25)*a(n-3) -4*(n-3)*(3*n^2-21*n+68)*a(n-4)=0. - _R. J. Mathar_, Jan 09 2020
%e a(8)=4 since we have the following four excursions of length 8: UHDHUHDH, HUHDHUHD, UHUHDHDH and HUHUHDHD.
%Y Cf. A000108.
%K nonn,walk
%O 0,5
%A _Valerie Roitner_, Nov 29 2019
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