A226995 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 the diagonal (if any) is an H step and the last step joining the diagonal (if any) is a S step.

1, 2, 6, 23, 103, 504, 2588, 13661, 73373, 398814, 2186818, 12072275, 67004451, 373532596, 2089994360, 11730304377, 66012996217, 372350924666, 2104523577534, 11916013288271, 67576932913951, 383781666337072, 2182362613988116, 12424357722805333

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: x/((x-1)^2*sqrt(x^2-6*x+1)) - 1/(x-1).

a(n) ~ (3+2*sqrt(2))^(n+1/2)/(2^(3+1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 27 2013

Example

a(0) = 1: the empty path.
a(1) = 2: HS, U.
a(2) = 6: HHSS, HSHS, HSU, HUS, UHS, UU.
a(3) = 23: HHHSSS, HHSHSS, HHSSHS, HHSSU, HHSUS, HHUSS, HSHHSS, HSHSHS, HSHSU, HSHUS, HSSHHS, HSUHS, HSUU, HUHSS, HUSHS, HUSU, HUUS, UHHSS, UHSHS, UHSU, UHUS, UUHS, UUU.

Maple

a:= proc(n) option remember; `if`(n<4, [1, 2, 6, 23][n+1],
     ((8*n-11)*a(n-1) +(21-14*n)*a(n-2)
      +(8*n-13)*a(n-3) -(n-2)*a(n-4))/ (n-1))
    end:
seq(a(n), n=0..25);

Mathematica

CoefficientList[Series[x/((x-1)^2*Sqrt[x^2-6*x+1]) - 1/(x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 27 2013 *)

Crossrefs

Cf. A001850 (unrestricted paths), A006318 (subdiagonal paths), A226994, A226996.

Sequence in context: A279573 A174193 A238639 * A301897 A374546 A022558

Adjacent sequences: A226992 A226993 A226994 * A226996 A226997 A226998

Keyword

nonn

Author

Alois P. Heinz, Jun 26 2013

Status

approved