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%I #4 Jul 26 2022 10:39:01
%S 1,0,1,0,2,0,5,1,14,7,43,36,143,166,509,731,1915,3158,7523,13560,
%T 30537,58257,127029,251266,538253,1089666,2313121,4754148,10051130,
%U 20868070,44065633,92132176,194617333,408971295,864899013,1824485600,3864369141
%N Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at odd level.
%C a(n)=A167634(n,0).
%F G.f.: G = [1 + 2z - z^3 - sqrt(1 - 4z^2 - 2z^3 + z^6)]/[2z(1 + z - z^2)].
%F D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+5)*a(n-2) +(-2*n+7)*a(n-3) +3*a(n-4) +a(n-5) +(n-6)*a(n-6)=0. - _R. J. Mathar_, Jul 26 2022
%e a(6)=5 because we have UUDDUUDDUUDD, UUDDUUUUDDDD, UUUUDDDDUUDD, UUUUDDUUDDDD, and UUUUUUDDDDDD.
%p G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z-z^2)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);
%Y Cf. A167634, A167638
%K nonn
%O 0,5
%A _Emeric Deutsch_, Nov 08 2009
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