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Search: id:A110107
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| A110107 |
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Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k return steps to the line y=x from the line y=x+1 or from the line y=x-1 (i.e. E steps from the line y=x+1 to the line y=x or N steps from the line y=x-1 to the line y=x; 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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| 1, 1, 2, 1, 8, 4, 1, 26, 28, 8, 1, 88, 136, 80, 16, 1, 330, 600, 512, 208, 32, 1, 1360, 2636, 2768, 1648, 512, 64, 1, 6002, 11892, 14024, 10544, 4832, 1216, 128, 1, 27760, 55376, 69728, 60768, 35712, 13312, 2816, 256, 1, 132690, 265200, 347072, 332768, 231232
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row sums are the central Delannoy numbers (A001850). sum(k*T(n,k),k=0..n)=2*A110099(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.=1/(1-z-2tzR), 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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T(2,1)=8 because we have DN(E), DE(N), N(E)D, ND(E), NNE(E), E(N)D, ED(N) and EEN(N) (the return E or N steps are shown between parentheses).
Triangle begins:
1;
1,2;
1,8,4;
1,26,28,8;
1,88,136,80,16;
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z-2*t*z*R): Gser:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001850, A110098, A110099.
Sequence in context: A085470 A099379 A133214 this_sequence A110446 A109979 A110171
Adjacent sequences: A110104 A110105 A110106 this_sequence A110108 A110109 A110110
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 11 2005
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