A243871 - OEIS

%I #12 Jun 13 2014 19:06:49

%S 1,3,10,35,124,454,1684,6305,23781,90209,343809,1315499,5050144,

%T 19442366,75034354,290203076,1124511549,4364693311,16966567970,

%U 66041815437,257378634365,1004167036295,3921726323436,15330264382726,59977821022143,234839855088313

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

%H Alois P. Heinz, <a href="/A243871/b243871.txt">Table of n, a(n) for n = 5..1000</a>

%F 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 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, 2)))

%p     end:

%p a:= n-> coeff(b(2*n, 0, 1), z, 1):

%p seq(a(n), n=5..40);

%Y Column k=1 of A243881.

%Y Column k=738 of A243827.

%K nonn

%O 5,2

%A _Alois P. Heinz_, Jun 13 2014