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A243871 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

%I A243871 #12 Jun 13 2014 19:06:49
%S A243871 1,3,10,35,124,454,1684,6305,23781,90209,343809,1315499,5050144,
%T A243871 19442366,75034354,290203076,1124511549,4364693311,16966567970,
%U A243871 66041815437,257378634365,1004167036295,3921726323436,15330264382726,59977821022143,234839855088313
%N A243871 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).
%H A243871 Alois P. Heinz, <a href="/A243871/b243871.txt">Table of n, a(n) for n = 5..1000</a>
%F A243871 a(n) = (2*(2*n-17) *(2*n-19) *(2*n-9) *a(n-1) -(2*n-19) *(6*n^2-75*n+208) *a(n-4) +2*(2*n-17) *(10*n^2-136*n+387) *a(n-5) -(2*n-19) *(6*n^2-75*n+212) *a(n-8) +(32*n^3-704*n^2+4940*n-10850) *a(n-9) -(2*n-17) *(2*n-9) *(n-14) *a(n-10) -(2*n-19) *(n-8) *(2*n-9) *a(n-12) +2*(2*n-9) *(2*n^2-36*n+161) *a(n-13) -(n-10) *(2*n-17) *(2*n-9) *a(n-14)) / ((2*n-17) *(2*n-19) *(n-4)).
%p A243871 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
%p A243871      series(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
%p A243871       z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]), z, 2)))
%p A243871     end:
%p A243871 a:= n-> coeff(b(2*n, 0, 1), z, 1):
%p A243871 seq(a(n), n=5..40);
%Y A243871 Column k=1 of A243881.
%Y A243871 Column k=738 of A243827.
%K A243871 nonn
%O A243871 5,2
%A A243871 _Alois P. Heinz_, Jun 13 2014