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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; 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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). sum(k*T(n,k),k=0..n-1)=2*A110127(n).
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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
| 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
| 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
| 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
| Cf. A001850, A001003, A100127, A006318, A110121.
Sequence in context: A098332 A096663 A133369 * A110221 A170856 A176781
Adjacent sequences: A110120 A110121 A110122 * A110124 A110125 A110126
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2005
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