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A110098
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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 (i.e. E 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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2
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1, 2, 1, 6, 6, 1, 22, 30, 10, 1, 90, 146, 70, 14, 1, 394, 714, 430, 126, 18, 1, 1806, 3534, 2490, 938, 198, 22, 1, 8558, 17718, 14002, 6314, 1734, 286, 26, 1, 41586, 89898, 77550, 40054, 13338, 2882, 390, 30, 1, 206098, 461010, 426150, 244790, 94554
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OFFSET
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0,2
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COMMENTS
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Row sums are the central Delannoy numbers (A001850). Column 0 yields the large Schroeder numbers (A006318). Column 1 yields A006320. Column k has g.f. z^k R^(2k+1), where R=1+zR+zR^2 is the g.f. of the large Schroeder numbers (A006318). sum(kT(n,k),k=0..n)=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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LINKS
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Table of n, a(n) for n=0..49.
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FORMULA
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T(n, k)=[(2k+1)/(n-k)]sum(binomial(n-k, j)*binomial(n+k+j, n-k-1), j=0..n-k) for k<n; T(n, n)=1; T(n, k)=0 for k>n. G.f.=R/(1-tzR^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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T(2,1)=6 because we have DN(E), N(E)D, N(E)EN, ND(E), NNE(E) and ENN(E) (the return E steps are shown between parentheses).
Triangle begins:
1;
2,1;
6,6,1;
22,30,10,1;
90,146,70,14,1;
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MAPLE
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T:=proc(n, k) if k=n then 1 else ((2*k+1)/(n-k))*sum(binomial(n-k, j)*binomial(n+k+j, n-k-1), j=0..n-k) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; #yields sequence in triangular form
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CROSSREFS
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Cf. A001850, A006318, A006320, A110099, A110107.
Sequence in context: A104684 A060538 A110183 * A130561 A157400 A091599
Adjacent sequences: A110095 A110096 A110097 * A110099 A110100 A110101
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Jul 11 2005
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STATUS
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approved
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