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 A110121 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). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 n contains 1 + floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). LINKS R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5. FORMULA T(n,0) = A110122(n). Sum_{k=0..floor(n/2)} k*T(n,k) = A110127(n). 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). EXAMPLE 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; MAPLE 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 MATHEMATICA 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}]] (* Jean-François Alcover, Dec 07 2011, after Maple *) CROSSREFS Cf. A001850, A006318, A110122, A110123, A110127. Sequence in context: A072117 A162853 A162854 * A288518 A069522 A170857 Adjacent sequences:  A110118 A110119 A110120 * A110122 A110123 A110124 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jul 13 2005 STATUS approved

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