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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).
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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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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).
The row sums are the central Delannoy numbers (A001850).
Column 0 yields the large Schroeder numbers (A006318).
Column k has g.f. z^k*R^(2*k+1), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
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LINKS
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FORMULA
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T(n,k) = ((2*k+1)/(n-k))*Sum_{j=0..n-k} binomial(n-k, j)*binomial(n+k+j, n-k-1) for k < n;
T(n,n) = 1;
T(n,k) = 0 for k > n.
G.f.: R/(1 - t*z*R^2), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
Sum_{k=0..n} k*T(n,k) = A110099(n).
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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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KEYWORD
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AUTHOR
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STATUS
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approved
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