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A243870 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

%I A243870 #14 Mar 28 2021 00:15:20
%S A243870 1,1,2,5,14,41,129,419,1395,4737,16338,57086,201642,718855,2583149,
%T A243870 9346594,34023934,124519805,457889432,1690971387,6268769864,
%U A243870 23320702586,87031840257,325741788736,1222429311437,4598725914380,17339388194985,65514945338284
%N A243870 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).
%C A243870 UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.
%H A243870 Alois P. Heinz, <a href="/A243870/b243870.txt">Table of n, a(n) for n = 0..1000</a>
%F A243870 Recursion: see Maple program.
%p A243870 a:= proc(n) option remember; `if`(n<14, [1, 1, 2, 5, 14, 41,
%p A243870        129, 419, 1395, 4737, 16338, 57086, 201642, 718855][n+1],
%p A243870        ((4*n-2)*a(n-1) -(3*n-9)*a(n-4) +(10*n-41)*a(n-5)
%p A243870        -(3*n-21)*a(n-8) +(8*n-64)*a(n-9) -(n-14)*a(n-10)
%p A243870        -(n-11)*a(n-12) +(2*n-25)*a(n-13) +(14-n)*a(n-14))/(n+1))
%p A243870     end:
%p A243870 seq(a(n), n=0..40);
%t A243870 a[n_] := a[n] = If[n<14, {1, 1, 2, 5, 14, 41, 129, 419, 1395, 4737, 16338, 57086, 201642, 718855}[[n+1]], ((4n-2)a[n-1] - (3n-9)a[n-4] + (10n-41)a[n-5] - (3n-21)a[n-8] + (8n-64)a[n-9] - (n-14)a[n-10] - (n-11)a[n-12] + (2n-25)a[n-13] + (14-n)a[n-14])/(n+1)];
%t A243870 a /@ Range[0, 40] (* _Jean-François Alcover_, Mar 27 2021, after _Alois P. Heinz_ *)
%Y A243870 Column k=0 of A243881.
%Y A243870 Column k=738 of A243753.
%K A243870 nonn
%O A243870 0,3
%A A243870 _Alois P. Heinz_, Jun 13 2014