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A110123
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Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).
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1
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1, 3, 11, 2, 45, 16, 2, 197, 100, 22, 2, 903, 576, 174, 28, 2, 4279, 3206, 1202, 266, 34, 2, 20793, 17568, 7732, 2128, 376, 40, 2, 103049, 95592, 47676, 15452, 3408, 504, 46, 2, 518859, 518720, 286156, 105528, 27500, 5096, 650, 52, 2, 2646723, 2813514
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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).
Row 0 has one term; row n has n terms (n > 0).
Row sums are the central Delannoy numbers (A001850).
Column 0 yields the little Schroeder numbers (A001003).
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LINKS
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FORMULA
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Sum_{k=0..n-1} k*T(n,k) = 2*A110127(n).
G.f.: (1 - tzR + zR)/(1 - z - tzR + tz^2*R - zR - z^2*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).
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EXAMPLE
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T(2,1)=2 because we have NEEN and ENNE.
Triangle begins:
1;
3;
11, 2;
45, 16, 2;
197, 100, 22, 2;
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=simplify((1-z*R*t+z*R)/(1-z-z*R*t+z^2*R*t-z*R-z^2*R)): Gser:=simplify(series(G, z=0, 14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: 1; for n from 1 to 10 do seq(coeff(t*P[n], t^k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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