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A110121
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Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k EE's crossing the line y=x (i.e. two consecutive E 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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3
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1, 3, 12, 1, 53, 10, 247, 73, 1, 1192, 474, 17, 5897, 2908, 183, 1, 29723, 17290, 1602, 24, 152020, 100891, 12475, 342, 1, 786733, 581814, 90205, 3780, 31, 4111295, 3329507, 620243, 35857, 550, 1, 21661168, 18956564, 4114406, 307192, 7351, 38
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OFFSET
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0,2
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COMMENTS
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Row n contains 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). T(n,0)=A110122(n). sum(k*T(n,k),k=0..floor(n/2))=A110127(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..41.
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FORMULA
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G.f.=1/[(1-zR)^2-z-tz^2*R^2], 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(2,0)=12 because, among the 13 (=A001850(2)) Delannoy paths of length 2, only NEEN has an EE crossing the line y=x.
Triangle begins:
1;
3;
12,1;
53,10;
247,73,1;
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/((1-z*R)^2-z-t*z^2*R^2): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 12 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form
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MATHEMATICA
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nmax = 11; r := (1 - z - Sqrt[1 - 6*z + z^2])/2/z; g := 1/((1 - z*r)^2 - z - t*z^2*r^2); gser = Series[g, {z, 0, nmax}]; p[0] = 1; Do[ p[n] = Coefficient[ gser, z, n] , {n, 1, nmax}]; row[n_] := Table[ Coefficient[ t*p[n], t, k], {k, 1, 1 + Floor[n/2]}]; Flatten[ Table[ row[n], {n, 0, nmax}]] (* From Jean-François Alcover, Dec 07 2011, after Maple *)
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CROSSREFS
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Cf. A006318, A001850, A110122, A110123, A110127.
Sequence in context: A072117 A162853 A162854 * A069522 A170857 A085296
Adjacent sequences: A110118 A110119 A110120 * A110122 A110123 A110124
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Jul 13 2005
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STATUS
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approved
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