A226996 - OEIS

%I #11 Jun 27 2013 16:15:05

%S 1,1,2,10,59,339,1908,10660,59493,332469,1861910,10451086,58793535,

%T 331434215,1871929768,10590886536,60014622089,340566437545,

%U 1935134951402,11008701669202,62694973984771,357406440776891,2039344466594972,11646264778160300,66561506740727149

%N Number of lattice paths from (0,0) to (n,n) consisting of steps U=(1,1), H=(1,0) and S=(0,1) such that the first step leaving and the last step joining the diagonal (if any) is an H step.

%H Alois P. Heinz, <a href="/A226996/b226996.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: sqrt(x^2-6*x+1)/(4*(x-1)^2)+1/(4*sqrt(x^2-6*x+1))-1/(2*(x-1)). - _Vaclav Kotesovec_, Jun 27 2013

%F a(n) ~ sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(16*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 27 2013

%e a(0) = 1: the empty path.

%e a(1) = 1: U.

%e a(2) = 2: HSSH, UU.

%e a(3) = 10: HHSSSH, HSHSSH, HSSHSH, HSSHU, HSSSHH, HSSUH, HSUSH, HUSSH, UHSSH, UUU.

%p a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 10][n+1],

%p      ((8*n^3-35*n^2+49*n-21)*a(n-1) -(2*n-3)*(7*n^2-21*n+15)*a(n-2)

%p       +(8*n^3-37*n^2+55*n-27)*a(n-3) -(n-3)*(n-1)^2*a(n-4))

%p      / (n*(n-2)^2))

%p     end:

%p seq(a(n), n=0..30);

%t CoefficientList[Series[Sqrt[x^2-6*x+1]/(4*(x-1)^2)+1/(4*Sqrt[x^2-6*x+1])-1/(2*(x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 27 2013 *)

%Y Cf. A001850 (unrestricted paths), A006318 (subdiagonal paths), A226994, A226995.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jun 26 2013