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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). 4
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
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
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
Sequence in context: A162853 A162854 A342787 * A321710 A358325 A288518
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 13 2005
STATUS
approved

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Last modified February 22 08:27 EST 2024. Contains 370240 sequences. (Running on oeis4.)