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A114580
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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k ascents (0<=k<=floor(n/2)); an ascent is a maximal string of upsteps.
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1
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1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 14, 6, 1, 26, 23, 1, 1, 46, 70, 10, 1, 79, 186, 56, 1, 1, 133, 451, 235, 15, 1, 221, 1025, 825, 115, 1, 1, 364, 2220, 2562, 630, 21, 1, 596, 4634, 7274, 2794, 211, 1, 1, 972, 9396, 19286, 10696, 1456, 28, 1, 1581, 18612, 48450, 36715
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OFFSET
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0,6
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COMMENTS
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Row n contains 1+floor(n/2) terms. Row sums are the Motzkin numbers (A001006). Sum(k*T(n,k),k=0..floor(n/2))=A005774(n-1).
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LINKS
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FORMULA
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G.f.: G(t,z) satisfies G = 1+zG+z^2[t(1+zG)+G-1-zG]G.
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EXAMPLE
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T(4,1) = 7 because we have HH(U)D, H(U)DH, H(U)HD, (U)DHH, (U)HDH, (U)HHD and (UU)HH, where U=(1,1), H=(1,0), D=(1,-1) (the ascents are shown between parentheses).
Triangle begins:
1;
1;
1, 1;
1, 3;
1, 7, 1;
1, 14, 6;
1, 26, 23, 1;
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MAPLE
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G:=1/2*(1-z+z^2-t*z^2-sqrt(1-z^2-2*z+2*z^3-2*z^3*t-2*z^2*t+z^4-2*z^4*t+z^4*t^2))/z^2/(z*t+1-z): Gserz:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 0 to 15 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, expand(`if`(t, 1, z)*b(x-1, y-1, true)
+b(x-1, y+1, false)+b(x-1, y, false))))
end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, false)):
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MATHEMATICA
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b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[If[t, 1, z]*b[x-1, y-1, True] + b[x-1, y+1, False] + b[x-1, y, False]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, False]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
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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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