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A243986 Number of Dyck paths of semilength n avoiding all five consecutive patterns of Dyck paths of semilength 3.

%I A243986 #18 Dec 21 2020 07:16:03
%S A243986 1,1,2,0,1,1,4,11,29,81,220,608,1676,4633,12847,35685,99367,277256,
%T A243986 775197,2171691,6095329,17138861,48274370,136197884,384868351,
%U A243986 1089211676,3087038820,8761410780,24898994687,70850054269,201848300443,575723018363,1643931888516
%N A243986 Number of Dyck paths of semilength n avoiding all five consecutive patterns of Dyck paths of semilength 3.
%C A243986 The consecutive patterns 101010, 101100, 110010, 110100, 111000 are avoided.  Here 1=Up=(1,1), 0=Down=(1,-1).
%H A243986 Alois P. Heinz, <a href="/A243986/b243986.txt">Table of n, a(n) for n = 0..1000</a>
%F A243986 Recurrence: see Maple program.
%F A243986 a(n) ~ c * d^n / n^(3/2), where d = 2.97831791935148503707065... is the root of the equation 4 + 12*d + 9*d^2 - 8*d^3 - 28*d^4 - 32*d^5 - 14*d^6 + 10*d^7 + 30*d^8 + 24*d^9 + 13*d^10 - 2*d^11 - 5*d^12 - 2*d^13 + d^14 = 0, c = 0.232860224447544532825428... . - _Vaclav Kotesovec_, Sep 06 2014
%e A243986 a(n) = A000108(n) for n<3.
%e A243986 a(3) = 0 because no Dyck path of semilength 3 can avoid itself.
%e A243986 a(4) = 1: 11001100.
%e A243986 a(5) = 1: 1110011000.
%e A243986 a(6) = 4: 101110011000, 110011001100, 111001100010, 111100110000.
%e A243986 a(7) = 11: 10101110011000, 10111001100010, 10111100110000, 11001110011000, 11011100110000, 11100110001010, 11100110001100, 11100110011000, 11110011000010, 11110011000100, 11111001100000.
%p A243986 a:= proc(n) option remember; `if`(n<18, [1$2, 2, 0, 1$2, 4, 11, 29,
%p A243986        81, 220, 608, 1676, 4633, 12847, 35685, 99367, 277256][n+1],
%p A243986       ((4*n-80)*a(n-18) +(16*n-302)*a(n-17) +(17*n-295)*a(n-16)
%p A243986       -(15*n-273)*a(n-15) -(61*n-971)*a(n-14) -(73*n-1043)*a(n-13)
%p A243986       -(19*n-191)*a(n-12) +(64*n-857)*a(n-11) +(114*n-1281)*a(n-10)
%p A243986       +(90*n-855)*a(n-9) +(11*n-40)*a(n-8) -(53*n-433)*a(n-7)
%p A243986       -(74*n-478)*a(n-6) -(42*n-225)*a(n-5) -(7*n-50)*a(n-4)
%p A243986       +(10*n-17)*a(n-3) +(6*n-12)*a(n-2) +(n-2)*a(n-1))/(n+1))
%p A243986     end:
%p A243986 seq(a(n), n=0..40);
%t A243986 b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[ Sum[b[x - 1, y - 1 + 2j, Mod[2t + j, 32]]*If[MemberQ[{42, 44, 50, 52, 56}, 2t + j], z, 1], {j, 0, 1}]]]];
%t A243986 a[n_] := Coefficient[b[2n, 0, 0], z, 0];
%t A243986 a /@ Range[0, 40] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ in A243998 *)
%Y A243986 Column k=0 of A243998.
%Y A243986 Cf. A000108, A014486, A063171, A243966.
%K A243986 nonn
%O A243986 0,3
%A A243986 _Alois P. Heinz_, Jun 16 2014