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A243873 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

%I A243873 #5 Jun 13 2014 16:46:43
%S A243873 1,5,23,105,472,2118,9446,41847,184256,806740,3514298,15238732,
%T A243873 65803650,283077978,1213561196,5186141801,22098720181,93913940321,
%U A243873 398127653185,1683928072645,7107304159469,29938529102885,125880340885997,528371537192555,2214227613719264
%N A243873 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).
%H A243873 Alois P. Heinz, <a href="/A243873/b243873.txt">Table of n, a(n) for n = 13..400</a>
%p A243873 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
%p A243873      series(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
%p A243873       z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]), z, 4)))
%p A243873     end:
%p A243873 a:= n-> coeff(b(2*n, 0, 1), z, 3):
%p A243873 seq(a(n), n=13..45);
%Y A243873 Column k=3 of A243881.
%K A243873 nonn
%O A243873 13,2
%A A243873 _Alois P. Heinz_, Jun 13 2014