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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..46.

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

FORMULA

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

EXAMPLE

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;

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

CROSSREFS

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

Cf. A005573, A001405, A000302.

Sequence in context: A188481 A138681 A038231 * A303054 A143496 A143697

Adjacent sequences:  A104852 A104853 A104854 * A104856 A104857 A104858

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 23 2005

STATUS

approved

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Last modified March 25 10:23 EDT 2019. Contains 321470 sequences. (Running on oeis4.)