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A129179
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Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2). A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis.
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1
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1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 4, 3, 2, 1, 1, 1, 1, 4, 6, 7, 10, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 1, 1, 1, 5, 10, 14, 21, 28, 31, 33, 34, 34, 31, 27, 25, 22, 17, 14, 13, 10, 7, 5, 4, 3, 2, 1, 1, 1, 1, 6, 15, 25, 40, 60, 77, 92, 106, 117, 122, 121, 120, 116, 107, 98, 91, 82, 71, 62, 54, 45
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OFFSET
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0,5
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COMMENTS
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Row n has 1+n^2 terms. Row sums are the large Schroeder numbers (A006318). Sum(k*T(n,k),k>=0)=A129180(n).
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LINKS
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Table of n, a(n) for n=0..82.
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FORMULA
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G.f.=G(t,z) satisfies G(t,z)=1+zG(t,z)+tzG(t,t^2*z)G(t,z).
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EXAMPLE
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T(3,5)=3 because we have UDUUDD, UUDDUD and UHHD.
Triangle starts:
1;
1,1;
1,2,1,1,1;
1,3,3,3,4,3,2,1,1,1;
1,4,6,7,10,11,10,9,8,7,5,4,3,2,1,1,1;
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MAPLE
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G:=1/(1-z-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i)*z-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G, z=0, 13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 6 do seq(coeff(P[n], t, j), j=0..n^2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A006318, A129180.
Sequence in context: A189006 A219924 A196929 * A120621 A201080 A039754
Adjacent sequences: A129176 A129177 A129178 * A129180 A129181 A129182
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Apr 08 2007
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STATUS
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approved
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