A297672 Array with four columns read by rows: T(n,k) = number of n step walks in the first octant on a square lattice with last step being right (k=1), left (k=2), up (k=3) or down (k=4).

1, 0, 0, 0, 1, 1, 1, 0, 3, 1, 1, 1, 6, 6, 6, 2, 20, 10, 10, 10, 50, 50, 50, 25, 175, 105, 105, 105, 490, 490, 490, 294, 1764, 1176, 1176, 1176, 5292, 5292, 5292, 3528, 19404, 13860, 13860

Offset: 1

Author

Roger Ford, Jan 02 2018

Keywords

• nonn
• tabf
• walk
• more

Comments

Sum_{k=1..4} T(n,k) = A005558(n).

For n >= 1, the ratio of the numbers of right or up last steps to left or down last steps is floor((n+2)/2): floor(n/2). - Roger Ford, Oct 28 2019

Examples

			k=    1    2    3    4    total
N   right left up   down  walks
1     1    0    0    0    =1
2     1    1    1    0    =3
3     3    1    1    1    =6
4     6    6    6    2    =20
There are 6 walks of 4 steps in the octant with the last step right. T(4,1)=6 RRRR, RRLR, RLRR, RUDR, RURR, RRUR.
		

Crossrefs

Cf. A005558, A001263.

Formula

n=1: T(1,1)=1, T(1,2)=0, T(1,3)=0, T(1,4)=0;

n>1: T(n,1) = C(n,floor(n/2))*C(n-1,floor((n-1)/2)) - C(n,floor((n-1)/2))*C(n-1,floor((n-2)/2));

T(n,2) = T(n,3) = C(n-1,floor(n/2)-1)*C(n,floor(n/2)-1)/floor(n/2);

n odd: T(n,4) = T(n,2);

n even: T(n,4) = T(n,2)*((n/2-1)/(n/2+1));

For n > 1, T(n,2) = T(n,3) = A001263(n,floor(n/2)).