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Search: id:A129182
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| A129182 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the area between the x-axis and the path is k (n>=0; 0<=k<=n^2). |
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3
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| 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 7, 0, 7, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 14, 0, 17, 0, 16, 0, 16, 0, 14, 0, 11, 0, 9, 0, 7, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,14
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COMMENT
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Row n has n^2 + 1 terms. Row sums are the Catalan numbers (A000108). Sum(k*T(n,k),k=0..n^2)=A008549(n).
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FORMULA
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G.f.=G(t,z) given by G(t,z)=1+tzG(t,t^2*z)G(t,z).
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EXAMPLE
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T(4,10)=3 because we have UDUUUDDD, UUUDDDUD and UUDUDUDD.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,2,0,1,0,1;
0,0,0,0,1,0,3,0,3,0,3,0,2,0,1,0,1;
0,0,0,0,0,1,0,4,0,6,0,7,0,7,0,5,0,5,0,3,0,2,0,1,0,1;
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MAPLE
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G:=1/(1-t*z*g[1]): for i from 1 to 11 do g[i]:=1/(1-t^(2*i+1)*z*g[i+1]) od: g[12]:=0: Gser:=simplify(series(G, z=0, 11)): for n from 0 to 7 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 7 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. A000108, A008549.
Sequence in context: A111594 A105348 A016406 this_sequence A116857 A158971 A121467
Adjacent sequences: A129179 A129180 A129181 this_sequence A129183 A129184 A129185
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 08 2007
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