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%I #27 Jul 24 2022 13:12:24
%S 1,1,2,3,6,10,19,33,62,110,205,368,683,1235,2286,4153,7674,13986,
%T 25813,47150,86949,159077,293176,537014,989267,1813659,3339940,
%U 6127355,11280954,20706414,38114723,69988457,128809594,236602202,435400253,799958582,1471961037
%N Number of dispersed Dyck paths of semilength n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no UUU's (U=(1,1)).
%C a(n) = A191518(n,0).
%H Alois P. Heinz, <a href="/A191519/b191519.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 2/(1-2*x-x^2+sqrt(1-2*x^2-3*x^4)).
%F a(n-1)=sum(m=floor((n+1)/2)..n, ((2*m-n)*sum(j=0..m, binomial(j,n+2*j-3*m)*binomial(m,j)))/m). [_Vladimir Kruchinin_, Mar 09 2013]
%F D-finite with recurrence (n+1)*a(n) +(-n-1)*a(n-1) +3*(-n+1)*a(n-2) +(n-5)*a(n-3) +(-n+11)*a(n-4) +(5*n-19)*a(n-5) +3*(n-5)*a(n-6) +3*(n-5)*a(n-7)=0. - _R. J. Mathar_, Jul 24 2022
%e a(6) = 19 because among the 20 (=A001405(6)) dispersed Dyck paths of length 6 only UUUDDD has a UUU (U=(1,1), D=(1,-1)).
%p g := 2/(1-2*z-z^2+sqrt(1-2*z^2-3*z^4)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 36);
%p # second Maple program:
%p a:= proc(n) option remember;
%p `if`(n<7, [1, 1, 2, 3, 6, 10, 19][n+1],
%p ((n+1)*a(n-1)+(3*n-3)*a(n-2)-(n-5)*a(n-3)
%p +(n-11)*a(n-4)-(5*n-19)*a(n-5)-(3*n-15)*a(n-6)
%p -(3*n-15)*a(n-7))/(n+1))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jun 02 2014
%t a[n_] := Sum[((2*m-n-1)*Sum[Binomial[j, n+2*j-3*m+1]*Binomial[m, j], {j, 0, m}])/m, {m, Floor[(n+2)/2], n+1}]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Mar 31 2015, after _Vladimir Kruchinin_ *)
%Y Cf. A001405, A191518.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 07 2011
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