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A114583
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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).
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2
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1, 1, 2, 3, 1, 7, 2, 15, 6, 36, 14, 1, 85, 39, 3, 209, 102, 12, 517, 280, 37, 1, 1303, 758, 123, 4, 3312, 2085, 381, 20, 8510, 5730, 1194, 76, 1, 22029, 15849, 3657, 295, 5, 57447, 43914, 11187, 1056, 30, 150709, 122090, 33903, 3734, 135, 1, 397569, 340104
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OFFSET
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0,3
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COMMENTS
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Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 1 yields A114584. Sum(k*T(n,k),k=0..floor(n/3))=A005717(n-2).
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LINKS
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FORMULA
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G.f.=G=G(t, z) satisfies G=1+zG+z^2*G(tz-z+G).
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EXAMPLE
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T(5,1)=6 because we have HH(UHD), UD(UHD), (UHD)HH, (UHD)UD, H(UHD)H and U(UHD)D, where U=(1,1),H=(1,0),D=(1,-1) (the UHD's are shown between parentheses).
Triangle begins:
1;
1;
2;
3, 1;
7, 2;
15, 6;
36, 14, 1;
...
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MAPLE
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G:=(1-z-t*z^3+z^3-sqrt((1-3*z+z^3-t*z^3)*(1+z+z^3-t*z^3)))/2/z^2: Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 17 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
`if`(x=0, 1, b(x-1, y, `if`(t=1, 2, 0))+b(x-1, y-1, 0)*
`if`(t=2, z, 1)+b(x-1, y+1, 1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
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MATHEMATICA
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CoefficientList[#, t]& /@ CoefficientList[(1 - z - t z^3 + z^3 - Sqrt[(1 - 3z + z^3 - t z^3)(1 + z + z^3 - t z^3)])/2/z^2 + O[z]^17, z] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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