A069971 Table by antidiagonals of variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -k for the first time.

0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 8, 8, 8, 0, 0, 20, 22, 22, 20, 0, 0, 40, 48, 48, 48, 40, 0, 0, 70, 90, 92, 92, 90, 70, 0, 0, 112, 152, 160, 160, 160, 152, 112, 0, 0, 168, 238, 258, 260, 260, 258, 238, 168, 0, 0, 240, 352, 392, 400, 400, 400, 392, 352, 240, 0, 0, 330, 498

Offset: 0

Author

Henry Bottomley, Apr 29 2002

Keywords

• nonn
• tabl

Comments

Expected time to reach one of the boundaries at +n or -k for the first time is n*k, i.e. A004247.

Examples

			Rows start 0,0,0,0,0,0,0,...; 0,0,2,8,20,40,70...; 0,2,8,22,48,90,152...; 0,8,22,48,92,160,258...; etc.
		

Formula

T(n, k) =nk(n^2+k^2-2)/3 =T(n+1, k-1)/2+T(n-1, k+1)/2+(n-k)^2 with T(n, 0)=T(0, k)=0. T(n, n)=n^2*(n^2-1)*2/3=8*A002415(n).