A104855 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of three-dimensional lattice walks consisting of n unit steps, each in one of the six coordinate directions, starting at the origin, never going below the horizontal plane and having k vertical steps.

1, 4, 1, 16, 8, 2, 64, 48, 24, 3, 256, 256, 192, 48, 6, 1024, 1280, 1280, 480, 120, 10, 4096, 6144, 7680, 3840, 1440, 240, 20, 16384, 28672, 43008, 26880, 13440, 3360, 560, 35, 65536, 131072, 229376, 172032, 107520, 35840, 8960, 1120, 70, 262144, 589824

Offset: 0

Author

Emeric Deutsch, Apr 23 2005

Keywords

• nonn
• tabl

Examples

			T(2,1)=8 because we have NU, SU, EU, WU, UN, US, UE and UW, where N=(0,1,0),S=(0,-1,0), E=(1,0,0),W=(-1,0,0), U=(0,0,1) and S=(0,0,-1).
Triangle begins:
   1;
   4,  1;
  16,  8,  2;
  64, 48, 24,  3;
		

Links

• J. Brawner, Three-Dimensional Lattice Walks in the Upper Half-Space: Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.

Crossrefs

Row sums yield A005573. T(n,n) = A001405(n), T(n,0) = A000302(n) (powers of 4).

Cf. A005573, A001405, A000302.

Programs

Maple

T:=(n,k)->binomial(n,k)*binomial(k,ceil(k/2))*4^(n-k): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Formula

T(n, k) = binomial(n, k)*binomial(k, ceiling(k/2))*4^(n-k).