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%I #13 Dec 27 2020 19:44:04
%S 1,4,16,65,263,1077,4419,18132,74368,304778,1247972,5105477,20867862,
%T 85219608,347724794,1417697157,5775652743,23512922998,95657223246,
%U 388912046916,1580241458120,6417249216667,26046042351889,105661066012240,428430870576913
%N Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).
%H Alois P. Heinz, <a href="/A243872/b243872.txt">Table of n, a(n) for n = 9..400</a>
%H Vaclav Kotesovec, <a href="/A243872/a243872.txt">Recurrence (of order 14)</a>
%F a(n) ~ c * d^n * sqrt(n), where d = 3.992152919721564592666177480042427843835641823811... is the root of equation 1 - 2*d + d^2 - 6*d^5 + 2*d^6 - 4*d^9 + d^10 = 0, and c = 0.00000109315704269290466088403991068... . - _Vaclav Kotesovec_, Jul 16 2014
%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
%p series(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
%p z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]), z, 3)))
%p end:
%p a:= n-> coeff(b(2*n, 0, 1), z, 2):
%p seq(a(n), n=9..40);
%t b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Series[
%t b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]]+If[t==10, z, 1]*
%t b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]], {z, 0, 3}]]];
%t a[n_] := Coefficient[b[2n, 0, 1], z, 2];
%t a /@ Range[9, 40] (* _Jean-François Alcover_, Dec 27 2020, after _Alois P. Heinz_ *)
%Y Column k=2 of A243881.
%Y Column k=738 of A243828.
%K nonn
%O 9,2
%A _Alois P. Heinz_, Jun 13 2014
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