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Search: id:A110099
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| A110099 |
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Number of return steps to the line y=x from the line y=x+1 (i.e. E steps from the line y=x+1 to the line y=x) in all Delannoy paths of length n (a Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1), and D=(1,1)). |
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+0
3
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| 0, 1, 8, 53, 332, 2029, 12236, 73193, 435480, 2581273, 15258256, 90005981, 530071076, 3117718213, 18318316948, 107537570513, 630844709168, 3698457841201, 21671720364056, 126932183197061, 743158103135484
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n)=sum(k*A110098(n,k),k=0..n).
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REFERENCES
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R. A. Sulanke, Objects counted by the central Delannoy numbers, J. of Integer Sequences, 6, 2003, Article 03.1.5.
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FORMULA
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G.f.=zR^3/(1-zR^2)^2, where R=1+zR+zR^2 is the g.f. of the large Schroeder numbers (A006318).
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EXAMPLE
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a(2)=8 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely DD, DN(E), DEN, N(E)D, N(E)N(E), N(E)EN, ND(E), NNE(E), END, ENN(E), ENEN, EDN, and EENN, one has alltogether 8 return steps to the line y=x from the line y=x+1 (shown between parentheses).
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R^3/(1-z*R^2)^2: Gser:=series(G, z=0, 30): 0, seq(coeff(Gser, z^n), n=1..24);
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CROSSREFS
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Cf. A006318, A001850, A110098, A110107.
Sequence in context: A130153 A116171 A099622 this_sequence A091870 A054418 A070928
Adjacent sequences: A110096 A110097 A110098 this_sequence A110100 A110101 A110102
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 11 2005
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