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A110169
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Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (1,1) steps (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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1
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1, 2, 1, 10, 2, 1, 50, 10, 2, 1, 258, 50, 10, 2, 1, 1362, 258, 50, 10, 2, 1, 7306, 1362, 258, 50, 10, 2, 1, 39650, 7306, 1362, 258, 50, 10, 2, 1, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 1196834, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 6634890, 1196834
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums are the central Delannoy numbers (A001850). Column 0 yields A110170 (first differences of the central Delannoy numbers). sum(k*T(n,k),k=0..n)=A089165(n-1) (n>=1; partial sums of the central Delannoy numbers).
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REFERENCES
| 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
| T(n, k)=A001850(n-k)-A001850(n-k-1) for k<n; T(n, n)=1. T(n, k)=P_{n-k}(3)-P_{n-k-1}(3) for k<n; T(n, n)=1, where P_j is j-th Legendre polynomial. G.f.=(1-z)/[(1-tz)sqrt(1-6z+z^2)]
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EXAMPLE
| T(3,2)=2 because we have DDNE and DDEN.
Triangle starts:
1;
2,1;
10,2,1;
50,10,2,1;
258,50,10,2,1;
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MAPLE
| with(orthopoly): S:=proc(n, k) if k<n then P(n-k, 3)-P(n-k-1, 3) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(S(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001850, A110170, A089165.
Sequence in context: A132995 A114692 A112691 * A144274 A144275 A011268
Adjacent sequences: A110166 A110167 A110168 * A110170 A110171 A110172
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2005
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