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A132893
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Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e. left factors of Motzkin paths) and having k peaks (i.e. UDs; (0<=k<=floor(n/2)).
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1
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1, 2, 4, 1, 9, 4, 21, 13, 1, 50, 40, 6, 121, 118, 27, 1, 296, 340, 106, 8, 730, 965, 381, 46, 1, 1812, 2708, 1296, 220, 10, 4521, 7535, 4241, 935, 70, 1, 11328, 20828, 13482, 3676, 395, 12, 28485, 57266, 41916, 13658, 1940, 99, 1
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OFFSET
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0,2
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COMMENTS
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Row n has 1+floor(n/2) terms. T(n,0)=A091964(n). Row sums yield A005773. Sum(k*T(n,k),k=0..floor(n/2))=A132894(n-1).
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LINKS
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Table of n, a(n) for n=0..48.
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FORMULA
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G.f.=G=G(t,z) satisfies z(1-3z+z^(2)-tz^(2))G^(2)+(1-3z+z^(2)-tz^(2))G-1=0 (see the Maple program for the explicit expression of G).
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EXAMPLE
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T(3,1)=4 because we have HUD, UDH, UDU and UUD.
Triangle starts:
1;
2;
4,1;
9,4;
21,13,1;
50,40,6;
121,118,27,1;
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MAPLE
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G:=((-1+3*z-z^2+t*z^2+sqrt((1+z+z^2-t*z^2)*(1-3*z+z^2-t*z^2)))*1/2)/(z*(1-3*z+z^2-t*z^2)): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A091964, A132894, A005773.
Sequence in context: A114489 A101974 A097607 * A163240 A091958 A116424
Adjacent sequences: A132890 A132891 A132892 * A132894 A132895 A132896
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Oct 08 2007
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STATUS
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approved
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