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Search: id:A104855
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| A104855 |
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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. |
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1
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| 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; listen)
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OFFSET
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0,2
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REFERENCES
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J. Brawner, Solution of Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.
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FORMULA
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T(n, k)=binom(n, k)binom(k, ceil(k/2))4^(n-k).
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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Row sums yield A005573. T(n, n)=A001405(n) T(n, 0)=A000302(n) (powers of 4).
Cf. A005573, A001405, A000302.
Sequence in context: A067425 A138681 A038231 this_sequence A143496 A143697 A117438
Adjacent sequences: A104852 A104853 A104854 this_sequence A104856 A104857 A104858
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 23 2005
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